Essays in Late Time Cosmology

Essays in late time cosmology

A dissertation presented by

Nicholas Park

to The Department of Physics in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

Abstract

This thesis is comprised of two studies. The ﬁrst focuses on cosmic strings, and the

signatures they would leave in the polarization of the CMB if they are found to exist.

During matter domination, cosmic strings will form wakes which will give rise to

direct B-mode polarization in the CMB. The B-mode polarization power spectrum

due to these wakes is found to be similar in shape and smaller in size than the

power spectrum found from gravitational lensing. This will make it diﬃcult to ﬁnd

signatures of cosmic strings when looking at the B-mode polarization power spectrum.

A more promising approach will be to search for cosmic string wakes in position space

using edge detection algorithms. The second study focuses on the backreaction of

cosmological perturbations and its eﬀect on various distance measures. The study

begins by developing a framework that can be used to average over every relevant

source and observer, where each observer sees the same value for some clock ﬁeld,

each source and observer pair are null separated, and each source is at a particular

redshift from the observer. If this framework is used to study distance measures,

one ﬁnds potentially important corrections to luminosity distance redshift relation.

At small redshifts (z . 0.2), this correction is related to redshift space distortions.

At larger redshifts (z > 0.5), this eﬀect is related to a lensing correction. The low redshift correction is unlikely to have major implications on cosmological parameter estimation, but the lensing correction could be more important.

iii Resum´e

Cette thèse est composée de deux études. La première se concentre sur les cordes cosmiques et les signatures qu’elles laisseraient dans la polarisation du CMB. Pen- dant la domination de la matière, les cordes cosmiques formeront des sillages, qui donneront lieu àune polarisation directe en mode B dans le CMB . Le spectre de puissance de polarisation de mode B, àcause de ces sillages, se trouve àêtre sem- blable en terme de forme et plus petit en terme de taille que le spectre de puissance qui résulte de la lentille gravitationnelle. Ainsi, il sera difficile de trouver des signatures de cordes cosmiques lors de l’observation du spectre de puissance de polarisation de mode B. Une approche plus prometteuse sera la recherche de sillages des cordes cosmiques dans l’espace de position en utilisant des algorithmes de détection de droites. La seconde étude porte sur la backreaction et son effet sur les diverses mesures de distance. L’étude commence par développer un cadre qui peut être utilisé en moyenne sur chaque source et observateur pertinent, lorsque chaque observateur voit la même valeur pour un champ d’horloge, lorsque chaque source et l’observateur paire sont nulle séparés, et lorsque chaque source est àun redshift particulier relatif

àl’observateur. Si ce cadre est utilisépour étudier les mesures de distance, on trouve des corrections potentiellements importantes àla distance de luminosityérelation de décalage vers le rouge. A` de petits décalages vers le rouge (z . 0.2), cette correction est liée aux redshift distorsions spatiales. A décalage vers le rouge de plus grandes

(z > 0.5), cet effet est liéàune correction de lentille. La correction de redshift faible est peu probable d’avoir des implications majeures sur cosmologique estimation des paramètres, mais la correction de lentille pourrait être plus important.

iv Contents

TitlePage...... i Abstract...... iii Resum´e ...... iv Contents...... v ListofFigures...... viii Acknowledgments...... x CitationstoPreviouslyPublishedWork ...... xi

Introduction 1

I PolarizationDuetoCosmicStrings 13

1 Introduction:Polarization Due to Cosmic String Wakes 14

2 Cosmic String Wakes 16

3 Polarization and Power Spectra 23 3.1 Polarization ...... 23 3.1.1 Mechanism Causing Linear Polarization in the Universe .... 26 3.2 Power Spectrum and the Flat Sky Approximation ...... 28

4 PolarizationSignalDuetoCosmicStringWakes 32 4.1 PowerSpectraDuetoCosmicStringWakes ...... 37 4.1.1 Polarization Power Spectrum Due to a Single Cosmic String Wake...... 39 4.1.2 Total Power Spectrum Due to All Cosmic String Wakes . . . . 39 4.2 Conclusions ...... 45

II Redshift Luminosity Relation, Curvature and Dark En-

v Contents vi

ergy 46

5 Introduction: Luminosity Distance Redshift Relation 47 5.1 FLRWUniverses ...... 47 5.1.1 StressEnergyTensor ...... 48 5.1.2 FriedmannEquations...... 49 5.2 StandardCandlesandTypeIASupernova ...... 50 5.3 LuminosityRedshiftRelation ...... 52 5.4 Luminosity Redshift Relation in a General Spacetime ...... 55 5.5 Dark Energy: How Exactly is it Constrained by Observations?.... 57 5.6 OverviewoftheProblem...... 58 5.7 Averaging in Gravity and Previous Work Relating to Backreaction and Expansion...... 60

6 Perturbing an FLRW Universe 70 6.1 SolvingfortheGeometry...... 71 6.1.1 Zeroth Order Flat FLRW Matter Dominated Universe . . . . 71 6.1.2 First Order Cosmological Perturbation Theory ...... 72 6.1.3 Second Order Cosmological Perturbation Theory ...... 76 6.2 ShapeofthePowerSpectrum ...... 77 6.3 Evolution of Radiation Temperature in a Perturbed Universe..... 82 6.4 Geodesics and Comoving Observers, Light Cones and Light Cone Angles 83 6.4.1 GeodesicLightConeGauge ...... 90

7 Jacobi Maps and the Area Distance in a Perturbed Universe 94

8 Averaging Techniques and Consequences at Late Times 99 8.1 EnsembleandGeometricAveraging ...... 100 8.2 EmbeddedSubspaces...... 101 8.3 RestrictedGeometricAveraging ...... 106 8.4 SpatialAveragingandClockFields ...... 107 8.4.1 BuchertAveraging ...... 108 8.5 LightConeAveraging ...... 109 8.6 CombinedSpatialandLightConeAveraging ...... 113 8.7 PerformingtheCalculation...... 117 8.7.1 LocalObservables...... 117 8.7.2 NonlocalObservables...... 118 8.7.3 PerturbationTheory ...... 123 8.7.4 AdiabaticandEntropyClocks ...... 131 Contents vii

9 Averaged Local Observables 134 9.1 EnergyDensities ...... 136 9.1.1 Integrated Power Spectra and IR Sensitivity ...... 137 9.2 Ergodicity ...... 139 9.3 ScalarSpatialCurvature ...... 143 9.4 ExpansionandDeceleration ...... 145 9.5 Results...... 148

10 Averaging Nonlocal Observables 149 10.1 Fractional Correction to the Area Distance ...... 150 10.2 ExpansionandDeceleration ...... 162 10.3LargeRedshift ...... 163 10.4Discussion ...... 171

11 Fitting the Data 174 11.1 Least Squares Fitting and Perturbation Theory ...... 175 11.2 SizeofBackreaction ...... 177 11.3SupernovaData...... 180 11.3.1 FittingPerturbativeExpressions ...... 180 11.3.2 Fitting Nonperturbative Expressions to estimate the size of the Corrections ...... 181 11.4Conclusions ...... 186

Bibliography 188 List of Figures

2.1 Diagram showing a possible scenario resulting in the formation of a cosmicstringloop...... 19 2.2 Cosmic string with a deﬁcit angle 4πGµ moving from left to right. . . 20

3.1 Linearly polarized photon with polarization angle α and intensity I.. 24 3.2 Background CMB quadrupole anisotropy causing linear polarization when scattered oﬀ free electrons via Thompson scattering...... 27 3.3 Ionizationhistoryoftheuniverse ...... 28

4.1 This figure shows the position space signal of the polarization of light from the CMB after it has passed through a cosmic string wake. The Length of each of the lines shows the magnitude of polarization, and the orientation of the lines shows to orientation of the polarization. We can see the cosmic string has moved from the upper right hand corner towardtolowerlefthandcorner...... 33 4.2 Space time diagram showing a wake formed at time tj crossing the past light cone at time t...... 38 4.3 Shape function as defined in equation 4.10 where [s] is plotted against s. 40 F 4.4 Estimated upper bound B B mode power spectrum due to cosmic stringwakes...... − 43 4.5 Current bounds as well as the expected B B powerspectrum. . . . 44 − 5.1 Figure showing the luminosity distance as a function of redshift for a flat FLRW universe containing a single perfect fluid with equation of state ω = 1, 0, 1 plotted in “dotted blue”, “solid green” and “dashedred”respectively.{ − } ...... 54 5.2 Figure showing the luminosity distance as a function of redshift for an FLRW universe containing matter with positive zero and negative spatial curvature plotted in “dotted blue”, “solid green” and “dashed red”respectively...... 55

viii List of Figures ix

6.1 Spacetime diagram showing surfaces of constant τ and ω as well as indicating one of the light cone angles. We can see that based on this diagram, using τ, ω and θa as a coordinate system would be convenient when studying observables on the past light cone...... 93

8.1 Light cone averaging as described in [76]. In the frame on the left the average is taken over all sources at a particular value of the Buchert time (τ) on the past light cone of a particular observer. In the frame on the right the average is taken over all sources at a particular redshift. The two diagrams show that these averaging surfaces are, in general, different. The solid curves show an unperturbed universe and thedashedcurvesshowaperturbeduniverse...... 110 8.2 This figure illustrates combined spatial and light cone averaging. The dotted line shows all observers who measure a particular value for the clock field, the solid lines shows that all sources related to a particular observer are on that observer’s past light cone, and finally the dashed line picks all sources at a particular redshift for each observer. . . . . 114

11.1 Figures showing the luminosity distance redshift relation, using the data from [143] in blue. A best fit line of the background to the data is shown in green. Perturbative corrections are shown in red, and 1σ uncertainty associated with cosmic variance is shown in orange. The figure on the left allows for nonzero spatial curvature and the figure on the right makes the assumption the spatial curvature is zero. .... 181 11.2 Figures showing the cosmic variance in the luminosity distance as a function of redshift, using the data from [143] fitted to equations (10.12,eq:fractionalvarareadistlargez), both with (left) and without (right) spatialcurvature...... 182 11.3 Figure showing the luminosity distance redshift relation, using the data from [143] in blue. A best fit line to the data is shown in red, and 1σ uncertainty associated with cosmic variance is shown in orange. . . . 183 11.4 Figure showing the 68% confidence ellipse and best fit for the Hubble parameter and matter content of the universe, using supernova data alone...... 184 11.5 Figure showing the cosmic variance in the luminosity distance as a function of redshift, using the data from [143] fitted to a polynomial oforder5...... 184 11.6 Figures showing the luminosity distance redshift relation, using the data from [143] reduced by a factor of q (q = 10 left and q = 100 right) multiplied by the cosmic variance. A best fit line shown in red, and 1σ uncertainty associated with cosmic variance is shown in orange. . . . 185 Acknowledgments

I would like to express my gratitude to my supervisor, Robert Brandenberger, whose expertise, understanding, and patience has helped me at times when I needed it most.

His knowledge of a wide variety of topics has been most impressive and helpful when

I have been stuck on a problem. I would also like to thank him for the patience he has shown me in particular during the writing of this thesis.

I would also like to thank Robert de Mello Koch, whose passion inspired me to pursue physics.

I would also like to thank all the people who I have met in the Physics Department, both past and present students, and in particular the Cosmology group at McGill.

A special thanks to Grant Salton who collaborated with me on the work on which part I is based.

On a personal note, there are a number of people who have helped me along this journey. In particular, my mother and father who have always been a rock to lean on. I would like to thank my extended family for all their support an encouragement, especially my cousin, Roy, who made Montreal feel a little less far from home. I also want to thank numerous friends both in Montreal and Johannesburg for all they have done for me. Finally, I would like to thank Susanna, who has given me endless support.

I would like to recognize the Department of Physics for ﬁnancial support throughout my PhD.

x Citations to Previously Published Work

Chapter 4 is new work, and is based on [34], in which the polarization signal due to cosmic string wakes was studied. In this work I collaborated with Robert Branden- berger and Grant Salton. Grant and I contributed equally to the calculations.

Chapter 8 is mostly motivated by the works of Gaspernini et al [20, 76, 77]. In their work, they suggested the average for scalar observables on subspaces deﬁned by scalar constraints (see equation (8.10)) as well as proposing the light cone averaging

(see equation (8.12)) as well as using a notation which motivated the perturbative notation used in section 8.7.3. The spatial averaging was motivated by the works of

Buchert et al [39, 40]. In this work, I combined the ideas of these authors into the idea of different clock fields used to define spacelike hypersurfaces including showing the choice of clocks for which spatial averaging coincides with the definition of spatial averaging used by Buchert. I have also extended the work of Gasperini et al to average observables not only over all sources by also over all observables. This chapter serves to introduce the formalism used in the remainder of this thesis.

Chapters 9 and 10 are new unpublished work, in which the formalism discussed in chapter 8 is applied in order to better understand the role of backreaction on the luminosity distance redshift relation. Chapter 11 is also new unpublished work, which fits particular data to the results from chapters 9 and 10. The idea of fitting the variance to the model as opposed to finding the best fit in the model is new.

Aside from the motivations from [20, 76, 77] and [39, 40] in chapter 8, the contents of chapters 8, 9, 10 and 11 are all my own work, and all calculations results and discussions are all my own work although I must give thanks to my supervisor Robert

Brandenberger for discussions along the way.

xi Introduction

The standard model (or concordance model) assumed in modern cosmology is a spatially flat Friedmann–Lematre–Robertson–Walker (FLRW) universe with small fluctuations which can be treated perturbatively. It comprises six parameters that are then fitted to data from a number of observations. Of these six parameters, three are related to the matter content of the universe (baryonic matter, nonbaryonic cold dark matter and dark energy)1, two are related to the size and shape of the primordial

ns 1 power spectrum of scalar ﬂuctuations (P [k] = A k − where the amplitude of φ k0 scalar ﬂuctuations is given by A and the scalar spectral index is given by ns), and one

is related to ionization of the baryons at late times (the reionization optical depth is

given by τ).

Radiation refers to the photons in the universe. In the early universe, during thermal

equilibrium, a blackbody spectrum of radiation emerged. After the radiation stopped

interacting with free electrons (last scattering), that blackbody spectrum cooled via

expansion of the universe. Baryonic matter was tightly coupled to the radiation in

the early universe. Once the matter fell out of equilibrium with the radiation, the

matter moved freely from the radiation in the universe. Nonbaryonic dark matter

(hereafter simply referred to as dark matter, as no confusion arises in this study)

does not interact with radiation, and as such has always moved freely. This dark

matter could be hot, which would allow for some thermal pressure, or it could be

cold, implying nearly negligible amounts of pressure. The formation of large scale

structure favors cold dark matter over warm dark matter [32]. Since neutrinos have

such low mass, their thermal energy was greater than their mass in the early universe,

1This is in addition to nearly negligible amounts of radiation (which is well known from the temperature of the CMB) and neutrinos (which are usually accounted for in extended models)

1 Introduction 2

and as such they behaved like a noninteracting radiation ﬁeld. The universe has since

cooled and the neutrinos now look like a noninteracting matter ﬁeld. Finally, unlike

dark matter, dark energy does not clump and remains nearly constant in time.

The parameters of the concordance model are then ﬁtted to data from a number

of observations. Most notably among these observations are the Cosmic Microwave

Background (CMB)2 [5, 89], the abundance of light nuclei[57], the supernova brightness redshift relation[134, 123], and the formation of large scale structure in the universe [9].

The power spectrum of temperature ﬂuctuations, as well as the polarization power spectrum of photons from the CMB (see [5, 89]), are used to constrain the concordance model. The position and relative height of the acoustic peaks in the CMB are used to constrain the parameters of the concordance model. In particular, the position of acoustic peaks helps determine the sound speed at the last scattering surface.

This determines the total matter content of the universe, as well as the luminosity distance3 to the last scattering surface. The relative height of the acoustic peaks

is sensitive to the relative abundance of baryonic and dark matter. Low multipole

moments in the CMB are sensitive to the spatial curvature, and dark energy via

the late time integrated Sachs-Wolfe eﬀect. Since this eﬀect tests the evolution of

the scalar ﬂuctuation, it breaks the degeneracy between dark energy and curvature

with the luminosity distance. Photons scattering oﬀ free electrons after reionization

will also have an effect on CMB data. This effect has a nearly constant effect for

2In particular the power spectrum of CMB temperature fluctuations and polarization. 3It is actually the area distance which is related to the luminosity distance via a factor of (1+z)2 via the Etherington reciprocity relation[71]. Introduction 3 high multipoles. At low multipoles this effect is not as straight forward. As such, it is degenerate with the late time integrated Sachs Wolfe effect[116] and the overall size of scalar fluctuations. We can argue that data from CMB observations gives tight bounds on the baryonic matter content, the dark matter content, the luminosity distance to the surface of last scattering, as well as scalar spectral index, if one considers the large multipole moments. The scalar fluctuation amplitude, dark energy fraction, curvature and reionization optical depth are all model dependent and sensitive to the low multipole moments of the CMB data, with curvature having the largest effect.

In order to get the correct abundance of light elements, one needs to have very speciﬁc conditions in the early universe [51]. While there are still some issues related to the

Lithium abundance in the universe, the abundance of light nuclei in the universe is in agreement with CMB data discussed above. Since the light elements were formed during radiation domination, these results are almost independent of dark matter and dark energy. As such, it places a bound on the baryonic matter content of the universe. This bound is independent of and in agreement with CMB data, and as such gives strong evidence that the baryonic matter content of the universe is well understood.

Galaxy surveys are being used to ﬁnd the power spectrum of the galaxy distribution in the universe. If we can use the galaxy distribution in the universe as a tracer for the matter distribution in the universe, we have then measured the matter power spectrum Pδ[k,z] as a function of the redshift from an earth based observer. The evolution of the matter power spectrum can be used to break the degeneracy between dark energy, curvature, and the luminosity distance, in the same way that the late Introduction 4

time integrated Sachs Wolfe eﬀect can. In principle, these galaxy surveys are an

extremely powerful tool. A major problem we face is mapping the galaxy power

spectrum to the matter power spectrum. This problem is referred to as “biasing”

[61, 70]. Since the surveys have only been completed for galaxies at redshifts z . 0.2

(as well as luminous red galaxies at redshifts z . 0.5), a common approximation technique is to assume that the power spectrum is a constant in that range rather than trying to trace the evolution. We ﬁnd that this observation places tight bounds on the total matter density (both baryonic and dark) of the universe, as well as on the amplitude of scalar perturbations. This observation is in good agreement with other observations.

Neutral Hydrogen emits (and absorbs) photons when an excited electron decays back to the ground state. Two important lines, namely the Lyman-α (Ly α) line and the − 21cm line occur when electrons in neutral Hydrogen transition between an excited state and the ground state. The more energetic Ly α line occurs when an electron − transitions between the ﬁrst excited state and the ground state. The 21cm signal is much less energetic as it is associated with an electron in the ground state changing spin alignment relative to the proton in the nucleus. The state with the proton and electron spins aligned anti-parallel has slightly less energy than the state where they are aligned parallel.

In observing the brightness (or absorption) of these lines as a function of redshift, one is able to build a three dimensional power spectrum of the excited (or neutral)

Hydrogen as a function of redshift. Since baryonic matter is mostly comprised of

Hydrogen, studying the 21cm and Ly α lines should give a good indication of the − Introduction 5

baryonic matter power spectrum as a function of redshift. The Ly α line should be − present in photon spectra emerging from just after last scattering, as well as since reionization. Although the 21cm line is weak, it should be visible throughout the dark ages, and should be able to ﬁll in the missing data from the Ly α data set. − Once biasing has been properly accounted for, the Ly α, 21cm, and galaxy surveys − should all be complimentary data.

Aside from those listed above, there is evidence of dark matter in a number of observations. When observing galaxies, one finds that stars that are far from the galactic center are moving much faster than would be expected based on the distribution of luminous matter. It seems that some amount of nonluminous (dark) matter must be contained inside the galaxy[141]. When strong gravitational lensing occurs, multiple images or smeared images of distant objects are observed due to light being deflected towards a gravitational lens. Since the amount of light deflected is related to the gravitational potential, we can find the mass contained in a particular lens. While most lenses are not strong enough to see multiple images, one can often find shape distortions. This is called weak gravitational lensing and, once again, it indicates that some additional non luminous matter is required in the universe. See [30, 139] for discussions of lensing in general, as well as the use of weak lensing in determining cosmological parameters in [112, 90]. Aside from a possible detection of dark matter by the DAMA experiment [26] (this result is contested by [60] among others), there has been been no direct detection of nonbaryonic dark matter4. In spite of this the cosmological evidence supporting its existence is compelling.

4See [74, 64, 27] for reviews. Introduction 6

Dark energy, however, is more diﬃcult to constrain. Since it is modeled as a cosmological constant, there can be no particle physics detection of dark energy, and as such one relies on cosmological observations. As has been argued above, there is a strong degeneracy between the expression for the luminosity distance, dark energy, and curvature in the universe. There are a number of theoretical explanations for dark energy5. The most common explanation is a cosmological constant (although the value of the cosmological constant seems unnaturally small [151] 6). Alternatives include attempts to modify gravity either by including extra scalar degrees of freedom or by changing the underlying theory of gravity. The most conservative explanation is that dark energy can be explained, at least partially, by properly accounting for structure in the universe.

As can be seen from the previous arguments, the current baryonic matter density, as well as the current dark matter density, seem to be tightly constrained by a number of diﬀerent observations. As such, these will not be disputed in this work. Dark energy and curvature, on the other hand, are (nearly) degenerate with the luminosity distance.

Aside from the success of the concordance model, there are a number of observations which it does not accurately account for. The ﬁrst of these is the production of

Lithium-7 during big bang neucleosynthesis [51]. We find the ΛCDM model predicts about 3 times more Lithium-7 than is actually observed. The Integrated Sachs-Wolfe effect has now been observed [3], although the observed amplitude of the ISW effect

5See [56] for a review. 6See [126] for an argument pertaining to the difference between a bare and a renormalized cosmological constant. Introduction 7 is larger than that expected from the ΛCDM model. There is some some conflict between the cosmological parameters from different observations. Notable among

1 1 these are the value of the Hubble constant, H0 = 100hkms− Mpc− . CMB data

[89, 4] generally favors a lower Hubble constant (h 0.68) than supernova data [136] ∼ (h 0.73)7. Baryon acoustic oscillations can be used to find expressions for the area ∼ distance and Hubble constant. Recently [62]8 claims that the area distance predicted at z =2.34is2.5σ lower than that predicted by the ΛCDM model9. Similarly different observations give different values for the Hubble constant.

Despite the success of the concordance model of cosmology, the wealth of data makes cosmology a great testing ground for fundamental physics. A number of particle physics models of the early universe undergo phase transitions which, if correct, would give rise to a number of topological defects. The (non)detection of these defects constrains physics beyond the standard model. Theories that give rise to domain walls and topological monopoles are tightly constrained. Cosmic strings are not as tightly constrained, and searching for evidence of them is important. Part I deals with cosmic strings and, in particular, using the polarization of the CMB to place bounds on the string tension(Gµ). It begins with an introduction and a discussion of motivations for studying the signatures of cosmic string wakes. It then brieﬂy reviews cosmic strings, polarization, and power spectrum. Using these tools, the signature of cosmic string wakes in the polarization power spectrum is calculated.

7 See [22] for more details regarding the determination of H0 in a universe containing inhomogeneities 8See [12] where a similar calculation was performed for smaller redshifts, ﬁnding good agreement with CMB results. 9Baryon acoustic oscillations, CMB data and supernova data have been combined in [14]. Introduction 8

The results of this study are summarized below.

A cosmic string wake is equally likely to produce E or B mode polarization. • This is related to the fact that the wakes are not correlated with the CMB

quadrupole moment.

There is no cross correlation between the E and B mode polarization due to • cosmic string wakes. Similarly, there is no correlation between temperature and

polarization.

The largest contribution to the polarization comes from wakes formed shortly • after equal matter and radiation and that cross the past light cone shortly after

last scattering.

The polarization power spectrum turns over at l 1000, which is related to the • ∼ scale of equal matter and radiation.

The magnitude of the polarization power spectrum due to cosmic string wakes is • smaller than the B-mode polarization due to lensing. This makes it very diﬃcult

to distinguish the contribution to the B-mode polarization power spectrum due

to cosmic strings above the background contribution from lensing using power

spectrum measurements alone.

Evidence of cosmic string wakes is more likely to be found in position space • B-mode maps than in the B-mode polarization power spectra.

Unlike baryonic matter and dark matter, dark energy is particularly mysterious, and theoretical physics does not, as yet, have an elegant solution to the problem. Since Introduction 9 dark energy, curvature, and the luminosity distance redshift relation are nearly degenerate, the major goal of Part II, is to study the luminosity distance redshift relation in an inhomogeneous universe and the consequences for the dark energy mystery. Chap- ter 5 serves to introduce the luminosity distance redshift relation and its central role in modern cosmological measurement, as well as to brieﬂy review previous work in this ﬁeld. In chapter 6, the geometry of a matter dominated universe, with perturbative corrections to the curvature and dark energy content, is developed. This chapter also discusses a number of observables related to geodesics, the geodesic light cone coordinate system (introduced in [76]), as well as corrections to the area distance.

Chapter 8 gives a detailed treatment of averaging in cosmology. It begins with a discussion of the diﬀerence between ensemble averaging (averaging over all possible statistically equivalent realizations of a universe) and “geometric” averaging (averaging scalar quantities over subspaces of a Riemannian manifold) and the importance of each. We then consider submanifolds embedded in manifolds, and ﬁnd a very general expression for geometric averaging (which is in agreement with [77]). This geometric average is then applied to spatial and light cone averaging, as well as to combined spatial and light cone averaging. We discuss the central role of a clock

ﬁeld when considering spatial averaging, and relate our approach to the averaging deﬁned by Buchert. Light cone averaging, as presented in [76], is then discussed10.

Once this is discussed, the spatial averaging and light cone averaging are combined.

Finally, the chapter highlights the procedure for performing these geometric averages perturbatively up to second order.

10We will suggest an improved treatment of redshift. Introduction 10

This approach diﬀers from many other attempts to understand the eﬀects of backre-

action and its importance when calculating the energy content of the universe.

It averages over both sources and observers. There are many formalisms for • averaging over observers (see, for example, the works of Buchert [39, 40, 38] as

well as works by [155, 49, 118, 120, 115, 52, 36] among others). There is at

least one formalism that averages over sources but not observers (see the work

by Gasperini and collaborators [76, 20, 23, 21])

The role of a clock field when considering spatial averaging is discussed. The • role of an entropy field in the long wavelength limit is specifically considered.

The observable of interest is the luminosity distance as opposed to the local • expansion rate of the universe (which has previously been assumed to be closely

related to the slope of the luminosity distance redshift relation). It should be

noted that this is also true of the formalism described in [76].

It is defined in terms of directly measurable observables. The formalism de- • scribed in [76] defines averaging in terms of difference in time as measured by

comoving geodesic observers11.

Finally, the formalism for averaging developed in chapter 8 is used to ﬁnd an expression for the average area distance (as a function of the average energy density, dark energy density, curvature, and backreaction terms) in chapters 9 and 10. In chapter

11, this corrected form for the average luminosity distance redshift relation is ﬁtted to the supernova data from [143].

11This subtle point is discussed in detail in section 8.5. Introduction 11

The results of this study are as follows:

The leading correction to the area distance is similar in size to the zeroth order • expression when perturbation theory is used. This indicates a breakdown in

perturbation theory. Since perturbation theory breaks down, either an alter-

native approximation technique or a full numerical calculation are needed to

perform the calculation.

Redshift space distortions dominate the perturbative correction to the luminos- • ity distance redshift relation at small redshift (z < 0.3).

A lensing correction dominates the perturbative correction at high redshifts • (z > 0.3). This correction does not seem to adjust the energy density of the

universe but does appear to lower the equation of state.

Diﬀerent observers will, on average, see diﬀerent expressions for the luminosity • distance redshift relation. If, however, the clock in not sensitive to short wave-

length ﬂuctuations (such as the temperature of the CMB as measured by an

observer, or the time as measured by a geodesic observer), the clock sensitive

corrections are much smaller than either the redshift space distortions or the

lensing corrections to the luminosity distance.

In general, the slope of the luminosity distance redshift relation is not inversely • proportional to the expansion rate of the universe. Neither of these expressions

are related to the slope of the luminosity distance redshift relation above the

scale of spatial homogeneity [138]. Introduction 12

Corrections that are sensitive to long wavelength ﬂuctuations require the use of • a nonadiabatic clock ﬁeld.

The eﬀects of properly accounting for backreaction might require a slightly • larger Hubble constant and a slightly smaller dark energy fraction than assuming

backreaction does not contribute. It is possible that the size of this correction

to the various cosmological parameters is smaller than the uncertainty in these

parameters. Part I

Polarization Due to Cosmic Strings

13 Chapter 1

Introduction:Polarization Due to

Cosmic String Wakes

This chapter seeks to introduce cosmic strings, and their importance in physics. In

this part of the thesis we will work in units with c = 1 unless otherwise explicitly stated.

A large number of particle physics models will give rise to various topological defects.

If nature is described by one of these models, then the corresponding topological defect will be formed in the early universe and persist until the present time. The detection (or nondetection) of these topological defects constrains particle physics beyond the standard model.

One example of such topological defects is cosmic strings that will form long thin stable overdensities. In studying the signatures of, and therefore the bounds on, cosmic strings, new constraints can be placed on models for particle physics beyond the standard model. Currently, the best limit on cosmic strings has been found from

14 Chapter 1: Introduction:Polarization Due to Cosmic String Wakes 15

CMB temperature maps [66, 5, 125, 28, 18, 121]. These constraints give rise to limits

on the string tension.

Cosmic strings that are formed during matter domination1 will persist, forming a wake as they move through the universe. Shortly after electrons have recombined with atoms, an excess ionization fraction will remain in cosmic string wakes. This will lead to signatures in the CMB temperature and polarization maps, and therefore in the associated power spectra.

Unlike Gaussian scalar ﬂuctuations produced in the early universe, cosmic string wakes can lead to direct B-mode polarization in the CMB. In what follows, we will show that the B-mode polarization power spectrum due to cosmic string wakes has a similar shape with smaller amplitude than the B-mode polarization caused by gravitational lensing. The position space signal of the polarization due to cosmic string wakes, on the other hand, will give competitive bounds [59], due to the distinct signature of wakes. Chapter 4 shows that it is unlikely that a signal will be found when searching the B-mode power spectrum.

1Dark matter wakes will also form from strings laid down earlier. Chapter 2

Cosmic String Wakes

This chapter will give a brief overview of the origin of cosmic string wakes, and is

based roughly on [149]. It does not introduce any new concepts.

Topological defects emerge in some particle physics models when certain types of

phase transitions occur. Whether topological defects form or not depends on the

topology of the vacuum manifold.

As a toy model, consider a real scalar ﬁeld eﬀective action ( ) at energy scale (Λ): L

1 = ∂µφ∂ φ m2[Λ]φ2 λ[Λ]φ4 L 2 µ − −

2 If one finds a phase transition at Λcrit defined by (m [Λcrit] = 0), with an effective

2 2 mass m [Λ > Λcrit] > 0 and m [Λ < Λcrit] < 0, then at energies lower than the phase transition, there will be two vacuum states into which the theory can settle. Since the potential is symmetric, neither of these vacuum states is preferred. Regions that are not causally connected could settle into either of these two vacuum states.

16 Chapter 2: Cosmic String Wakes 17

Since both regions are in the vacuum state, there is no preference for one region over the other, and a stable discontinuity forms between the two regions. Since this discontinuity is in neither of the two vacuum states, it must contain some excess energy. This sort of discontinuity is referred to as a topological defect, and the mechanism by which it is formed is called the Kibble mechanism [99].

Since the universe continues to cool after the topological defect is formed, thermal

fluctuations will be insufficient to connect the different vacuum states and, as such, the topological defect will persist.

In general, diﬀerent topologies of the vacuum manifold after the phase transition will give rise to diﬀerent types of topological defects. In a (3 + 1) dimensional universe, if the vacuum manifold is a number of distinct points, such as described in the toy model, two dimensional topological defects called domain walls will form. If the vacuum manifold formed is isomorphic to U(1), then one-dimensional objects called cosmic strings will form. If the vacuum manifold formed is isomorphic to SU(2), then zero-dimensional objects called cosmic monopoles will form. Domain walls and cosmic monopoles are very tightly constrained by various experiments. Cosmic strings may still be viable, however. If they exist, they will give rise to a number of interesting observable signatures in the universe.

Long cosmic strings will form with a curvature radius similar to the Hubble length at the time of formation. These long cosmic strings will typically form a random walk and the segments will move with speeds close to the speed of light. This network of cosmic strings will have a complicated evolution, and occasionally long strings will intersect each other and form cosmic string loops. Figure 2.1 shows an example of Chapter 2: Cosmic String Wakes 18

a mechanism by which cosmic string loops may be formed when two long strings

intersect each other. These cosmic string loops will then radiate their energy via

gravitational radiation. This shows how the cosmic string network can lose energy.

The rate of energy lost by cosmic strings via this mechanism is proportional to the

correlation length (which is the comoving Hubble length at the time the string was

formed) of strings and the number of strings. Similarly, the rate that energy is trapped

in topological defects is proportional to the length of the string and the rate of string

formation. One can see then that the fraction of energy contained in cosmic strings

will approach a “scaling solution” [25, 122, 11], where the fraction of energy contained

in the cosmic string network is a constant.

Since the cosmic string loops decay, in this study the network of cosmic strings will

be modeled as an assembly of straight string segments with a physical length scale at

time t of c z[t]+1 , where c is a parameter that is best found from simulations. 1 (z[tj ]+1)H[tj ] 1

The parameters t and tj are the time of observation and time of formation respectively.

Cosmic Stings will typically move with speeds (vs) comparable to the speed of light,

and there are typically (1) cosmic strings per Hubble volume. Cosmic strings will O z[t]+1 typically travel a physical distance vsγs until they interact with another ∼ (z[tj ]+1)H[tj ]

string. We have deﬁned the speed of the string as vs, the Lorentz factor γs, the time of formation tj and the time of observation t.

Consider a cosmic string in its own rest frame in a Minkowski universe up to linear order in perturbation theory. Without loss of generality, one can choose to place the string along the z-direction. The metric and stress energy tensor can be written as1:

1See [149] for more details. Chapter 2: Cosmic String Wakes 19

Figure 2.1: Diagram showing a possible scenario resulting in the formation of a cosmic string loop.

gµν =ηµν + hµν

ν Tµ =µδ[x]δ[y]diag(1, 0, 0, 1)

Here µ is the mass per unit length of the string. Solving the Einstein equation, one can write the metric of space time as:

ds2 = dt2 + dz2 + dr2 + r2dθ¯2 − where θ¯ (0, 2π(1 2Gµ)] ∈ −

This metric looks like the usual ﬂat space metric except that space is covered by θ¯ Chapter 2: Cosmic String Wakes 20

which has a domain of less than 2π radians, or alternatively, space is a cone with deﬁcit angle 4πGµ (see ﬁgure 2.2).

Figure 2.2: Cosmic string with a deﬁcit angle 4πGµ moving from left to right.

Since a cosmic string will cause a deﬁcit angle, a cosmic string moving through the

background FLRW universe will cause energy behind the string to get kicked back to-

wards the line along which the cosmic string is moving, with a velocity δv =4πGµvsγs.

This will form an overdense region. During matter domination, this overdensity be-

hind the string will then further accrete matter, forming an overdense wake behind

the cosmic string.

Consider a particle which was initially at a comoving height q above the wake and

follows the evolution of the physical height h[q,t] as a function of time. For any

q>q∗[t] the expansion of space time wins over the kick back towards the wake and

any q

moving. q [t] can then serve as the deﬁnition of the comoving height of the wake. ∗ Chapter 2: Cosmic String Wakes 21

Deﬁning Ψ as the comoving displacement induced by gravitational accretion onto the

wake, we can write:

h[q,t]=a[t](q Ψ[q,t]) − ∂ h[q∗,t] =0 ∂t

Since we are considering a week ﬁeld limit, we can approximate the height of the

wake in the Newtonian limit. Introducing the gravitational potential Φ, and using

Newtons law, we have:

∂2 ∂ h[q,t]= Φ ∂t2 ∂h

The gravitational potential can be calculated using the surface tension (σ[t]) of the wake:

∂2 Φ =4πG (ρ + σδ[h]) ∂h2 t 2/3 σ(t) =4πGµt v γ ρ[t] i s s t j Solving the above diﬀerential equations, one ﬁnds that the physical height is given by2:

1/2 32π z[tj]+1 1 h[q∗[t],t] Gµv γ (2.1) ∼ 15 s s z[t]+1 H[t]

2See [142] for further details as well as numeric comparisons. Chapter 2: Cosmic String Wakes 22

A cosmic string laid down at time tj and observed at time t will form a wake of dimension:

z[t]+1 z[t]+1 32π z[t ]+1 1/2 1 c v γ Gµv γ j (2.2) 1 (z[t ]+1)H[t ] × s s (z[t ]+1)H[t ] × 15 s s z[t]+1 H[t] j j j j Since cosmic strings obey a scaling solution [25, 11], they will be continuously laid down for all times after the symmetry breaking scale. There will be an equal number3

(nH . 10) of wakes laid down per Hubble volume in each Hubble time.

3This number must be found from simulations. Chapter 3

Polarization and Power Spectra

In this chapter, the physics of polarization is brieﬂy reviewed. After that, power spectra and the ﬂat sky approximation are discussed. No new concepts are introduced in this chapter.

3.1 Polarization

Consider a photon moving out of the page as shown in ﬁgure 3.1. The electric ﬁeld will point along the dashed line and will have a maximum magnitude proportional to

I. We can write all the information in terms of the following four parameters:

I =I

Q =I cos[2α]

U =I sin[2α]

V =0

23 Chapter 3: Polarization and Power Spectra 24

Figure 3.1: Linearly polarized photon with polarization angle α and intensity I.

Here the Stokes parameters I,Q,U,V are introduced. If we consider a superposition

of photons, as are observed when studying the CMB, the same approach can be used

to decompose the incoming radiation.

I is the total intensity of the incoming light. If there is an overall polarization, it

can be characterized by the electric ﬁeld. One can deﬁne the angle between some

reference direction and the electric ﬁeld as α1. The magnitude of the polarization is

given by P I is the average magnitude of the electric ﬁeld2. The Stokes parameters ≤ Q and U are deﬁned as Q = P cos[2α] and U = P sin[2α]. Finally V is associated

with circular polarization, which is not relevant in this study.

With these deﬁnitions we can deﬁne the polarization tensor:

1α is an angle defined with respect to some arbitrary direction and, as such, is not an invariant quantity. 2If the superposition of photons is completely polarized, one will find that the photons all have their electric field propagating in the same direction, and P = I. If the superposition of photons is completely unpolarized, there will be no net electric field and P = 0. Chapter 3: Polarization and Power Spectra 25

P = E E αβ h α βi =Iδ + + ǫ V (3.1) αβ Pαβ αβ

Here is a time average of the incoming photons, where the average is taken over hi times longer than the period of the photon. The polarization tensor has been decom- posed into a trace part (I corresponding in cosmological terms to the temperature of the CMB), an antisymmetric part corresponding to circular polarization (V ) and a trace free symmetric tensor corresponding to linear polarization, ( ). We can write P the linear polarization tensor as:

1 Q U sin[θ] = − Pαβ 2   U sin[θ] Q sin2[θ]  − −    Since Q and U are deﬁned in terms of the angle α, they are not invariant quantities.

If one rotates the co-ordinate system, the Stokes parameters transform as:

Q¯ = Q cos[2α]+ U sin[2α]

U¯ = U cos[2α] Q sin[2α] −

Since the Stokes parameters associated with linear polarization are not invariant

quantities, it makes sense to search for alternative objects to study. We usually

consider the scalar pseudo-scalar decomposition [44]. In this case we can write: Chapter 3: Polarization and Power Spectra 26

2E = ∇ ∇α∇βPαβ 2B =ǫ (3.2) ∇ αβ∇γ∇βPαγ

The quantities E and B are now invariant under rotations, and are good observables.

3.1.1 Mechanism Causing Linear Polarization in the Uni-

verse

Consider the hot unpolarized photons moving along the positive y direction, and the

cold unpolarized photons moving in the negative z direction (as shown in ﬁgure 3.2)3.

If we consider the scattered photons to be moving in the positive x direction with

electric ﬁeld propagating in the y z plane, the z component of the scattered electric − ﬁeld can only come from hot photons and the y component of the scattered electric

ﬁeld can only come from the cold photons, there will be a net polarization along the z axis. This argument applies irrespective of the angles between the incoming and scattered photons, with some partial cancellation.

Therefore, a quadrupole anisotropy in the CMB will give rise to some linear polarization when it shines through an excess of free electrons. Since this eﬀect relies on the number of free electrons, it will depend on the ionization fraction of the universe. Figure 3.3 4 shows the approximate average ionization history of the universe.

A region of excess density will retain an excess of free electrons compared to the

3Figure 3.2 is taken from [91]. 4Figure 3.3 taken from [98]. Chapter 3: Polarization and Power Spectra 27

Quadrupole Anisotropy ε'

Thomson Scattering e– ε' ε

Linear Polarization

Figure 3.2: Background CMB quadrupole anisotropy causing linear polarization when scattered oﬀ free electrons via Thompson scattering. background for a short period of time. Since the universe remained ionized up to last scattering, any polarized photons from before then will scatter oﬀ free electrons, leaving little polarization. Most polarization in the CMB can be attributed to the

CMB quadrupole shining through free electrons shortly after last scattering, or since reionization, when the ionization fraction was largest. Chapter 3: Polarization and Power Spectra 28

100

xe 10−1 H /n

e −2

n 10

1+ze 10−3 1+zH

10−4 1 10 100 1000 1+z

Figure 3.3: Ionization history of the universe

3.2 Power Spectrum and the Flat Sky Approxima-

tion

If we consider two points in the sky, (namely ~x and ~y), we can project these two points onto a celestial sphere centered at an observer situated at ~0. We might consider the correlation function of some observable f, and since f is a function on the celestial sphere, it can be expanded in spherical harmonics (Ylm[θ,φ]= Ylm[Ω]).

f[Ω] = almYlm[Ω] Xl,m The correlation of f between ~x and ~y projected onto the celestial sphere can be written in terms of the angular power spectrum: Chapter 3: Polarization and Power Spectra 29

f[Ω]f[Ω′] = almal†′m′ Ylm[Ω]Yl†′,m′ [Ω′] h i ′ ′ l,lX,m,m D E f ′ ′ = Cl δll δmm Ylm[Ω]Yl†′m′ [Ω′] ′ ′ l,lX,m,m 2l +1 ~x ~y = Cf P · (3.3) 4π l l x y Xl | || |

Here, the usual deﬁnition for the angular power spectrum (Cl) is used in the ﬁrst line.

The assumption that the power spectrum is invariant under rotations and translations is used in the second line. The usual deﬁnition of the Legendre polynomial (Pl[x]) is

used in the ﬁnal step.

If we presume that the points ~x and ~y have a suﬃciently small angular separation

(~x ~y x y ), then we can project these vectors onto a ﬂat two dimensional plane, · ≈| || | tangential to the celestial sphere near ~x and ~y. In this case we can write f as a function of the coordinates on the plane (x) as f[x], which can be expanded as a

Fourier transform of f is then given by:

2 d k ik x f[x]= f¯[k]e · 2π Z In this case, we can make use of the ﬂat space form of the power spectrum: Chapter 3: Polarization and Power Spectra 30

assumption that the power spectrum is invariant under rotations and translations

is used in the second line. The usual deﬁnition of the Bessel function (j0[x]) is used

in the ﬁnal step.

Taking ~x = ~y, and equating these two expressions, we have:

2l +1 dk Cf = P [k] (3.5) 4π l k f Xl Z We can approximate the integral as a Riemann sum:

dk P [k] k f Z ∞ Pf [kl] = lim (kl kl 1) kl kl−1 0 k − − | − |→ l Xl=0 Pf [l] ≈ l Xl Using this approximation, we can show that: Chapter 3: Polarization and Power Spectra 31

2l +1 P [l] Cf f 4π l ≈ l Xl Xl Hence if l 1 ≫ l2 then Cf P [l] (3.6) 2π l ≈ f

f This is the flat sky approximation. It is easier to find Pf [l] than Cl since finding

f Pf [l] involves working exponential functions and finding Cl involves working with spherical harmonics, it is easier to find the angular power spectrum by using the flat

sky approximation, than ﬁnding it directly. We can write approximate the angular

power spectrum as:

f 2 f C0 k Pf [l] Cl lim (3.7) k 0 2 ≈ → Pf [k]! l

Here, the magnitude of Cl has been ﬁxed at l = 0 using the standard deﬁnition of

C0. The shape of the power spectrum has then been calculated using the ﬂat sky approximation:

2 2 1 d Ωd Ω′ f[Ω]f[Ω′] ′ ′ f 2 2 il(cos[θ](x1 x1)+sin[θ](x2 x2)) C = h i dθ d xd x′e − − f[x]f[x′] l 4π 2π d2xd2x f[x]f[x ] h i R ′ h ′ i Z Z 1 ′ ′ 2 2 il(cos[θ](x1 x1)+sin[θ](x2 x2)) = Rdθ d xd x′e − − f[x]f[x′] (3.8) 8π2 h i Z Z Chapter 4

Polarization Signal Due to Cosmic

String Wakes

This chapter begins with a summary of the results of [59], in which the position space signal of the linear polarization due to cosmic string wakes is described. The calculation of the scalar pseudo scalar decomposition as well as the calculation of the power spectra is novel, as was described in [34].

Since cosmic string wakes are eﬀectively matter overdensities, there will be excess polarization coming from them. In [59], the authors argue that a cosmic string wake will produce a linear polarization pattern that looks similar to ﬁgure 4.1, where one can clearly see a wake caused by a string moving from the top right to the bottom left.

The shape of the pattern is related to the orientation of the wake with respect to the observer. A cosmic string moving directly towards, or oriented towards, the observer will have a long thin signature whereas a string moving transverse and orientated transverse to the observer will have a rectangular signature.

32 Chapter 4: Polarization Signal Due to Cosmic String Wakes 33

Figure 4.1: This ﬁgure shows the position space signal of the polarization of light from

the CMB after it has passed through a cosmic string wake. The Length of each of

the lines shows the magnitude of polarization, and the orientation of the lines shows

to orientation of the polarization. We can see the cosmic string has moved from the

upper right hand corner toward to lower left hand corner.

As a ﬁrst approximation, we can assume that there is a wake laid down by a string

moving transverse to the line of sight between the observer and the string, with

orientation also transverse to the line of sight between the string and the observer.

This wake will then cause a rectangular pattern in the sky. We can then ﬁx the

co-ordinate system such that the wake is moving along the equator starting at φ =0

and extending to φ = δφ, and is extending from the equator to θ = π δθ. Here, δφ 2 − and δθ are related to the time at which the wake was created, and the time at which Chapter 4: Polarization Signal Due to Cosmic String Wakes 34

the wake crossed the past light cone. We can also introduce an overall constant (ρP )

related to the magnitude of polarization due to the wake [59]. A polarization angle α

related to the CMB quadrupole moment will also be associated with the wake. Since

the orientation of the wake is independent of the CMB quadrupole, this angle can be

treated as being arbitrary.

If the wake is suﬃciently small in the sky (δφ 1 and δθ 1), then we can write: ≪ ≪

1 φ π π ρ Θ[φ]Θ[δφ φ]Θ θ + δθ Θ θ Pαβ ≈2 p δφ − − 2 2 − h i h i cos[2α] sin[2α] −   (4.1) sin[2α] cos[2α]  − −    The expression depends on the time that the cosmic string was laid down (tj) as well as the time that the wake crosses the past light cone of the observer (t). The overall

1 amplitude of polarization (ρP ) has been factored out and is proportional to the CMB

quadrupole moment Q, to the Thompson cross section σT , to the free electron density

in the wake (ne[t,tj]), and to the integral along a null geodesic across the wake with

a height h[t,tj]. Although a slitly diﬀerent notation, this expression was derived in

[59]. The angular size of the wake is related to the size of the wake at the time it

crossed the past light cone.

1Although a slightly diﬀerent notation is used here, this expression was derived in [59]. Chapter 4: Polarization Signal Due to Cosmic String Wakes 35

1 3 t 2/3 ρ [t,t ]= Qσ n [t,t ]h[t,t ] 0 (4.2) p j 5 4π T e j j t r ρB[t] ΩB ne[t,tj] f f 2 ≈ mP ∼ 6πGmP t 24π t4 1/3 h[t,t ] Gµv γ j ≈ 5 s s t j 3 2 1/2 ρ 1.6 10− fGµQ(z[t]+1) (z[t ]+1) (4.3) p ∼ × j

Here, ne is simply the ionization fraction (f)times the number of atoms per unit volume2. The height of the wake was discussed in chapter (2) ad was given in equation (2.1). Using the dimensions of the wake given in equation (2.2), the angular sizes of the cosmic string wakes are then given by:

1 t0 − z[t]+1 t dτ(z[τ]+1) δφ arctan vsγs ≈  (z[tj]+1)H[tj] R z[t]+1 !   1  t0 − z[t]+1 t dτ(z[τ]+1) δθ = arctan c1  (z[tj]+1)H[tj] R z[t]+1 !    Assuming c 1 and v γ 1 1 ∼ s s ∼ 1/3 tj δθ = δφ ∆ arctan 1/3 ≡ ∼ "2(t t1/3)# 0 − 1 1/2 ∆ (z[t ]+1)− (4.4) ∼2 j

Ensemble averaging should technically involve an average over all orientations of the wake with respect to the observer, as well as overall polarization angles (α). Ulti- mately, this study ﬁnds that the signature of cosmic string wakes in CMB polarization

2Where we have assumed that all atoms are Hydrogen atoms Chapter 4: Polarization Signal Due to Cosmic String Wakes 36

is small in comparison to the polarization due to lensing. The averaging over wake

orientations leads to complicated corrections, which are even smaller. Since the aver-

age over orientations leads to such a small complicated eﬀect, it will not be considered

in what follows.

Averaging over the polarization angle leads to calculations of the form:

2π dα 1 cos2[2α]= 2π 2 Z0 2π dα 1 sin2[2α]= 2π 2 Z0 2π dα cos[2θ] sin[2α]=0 2π Z0 2π dα cos[2α]=0 2π Z0 2π dα sin[2α]=0 (4.5) 2π Z0 Averaging over the polarization angle and making use of the above expressions, as well as using equation 4.5, we can show that:

1 EE = BB = αβ h i h i 4 P Pαβ

EB = ET = BT =0 (4.6) h i h i h i

In the second line above, we have introduced the CMB temperature (T ). Since is

does not depend on the polarization angle α, the temperature and polarization are

uncorrelated.

Labeling each wake by capital Latin characters (e.g. I), and presuming diﬀerent

wakes are uncorrelated, it follows that: Chapter 4: Polarization Signal Due to Cosmic String Wakes 37

EI EJ = BI BJ 1 =δIJ 2 4 PI

EI BJ = EI T J = BI T J =0 (4.7)

The two important conclusions at this point are that cosmic string wakes give rise to

equal contribution to the B/B mode and E/E mode polarization, and do not give rise to any E/B polarization.

As a result we would expect to be able to ﬁnd the similar signatures from cosmic string wakes in both the E mode maps and the B mode maps.

This result is in contrast to the ﬂuctuations due to primordial scalar ﬂuctuations where:

ET =0 h i 6 EE > BB h i h i

Since B modes do not arise due to primordial scalar perturbations (at leading order), it makes more sense to search the B mode maps for signatures of cosmic string wakes.

4.1 Power Spectra Due to Cosmic String Wakes

The goal of this study is to ﬁnd the contribution to the polarization power spectra due to cosmic string wakes. Consider a wake that forms at time tj >teq and crosses the past light cone at time t>tls as shown in ﬁgure 4.2. The overdensity in the wake Chapter 4: Polarization Signal Due to Cosmic String Wakes 38

will lead to excess polarization. Since cosmic strings evolve according to a random

walk, we can assume that diﬀerent cosmic string wakes are uncorrelated, and therefore

that the polarization signal due to diﬀerent cosmic string wakes is uncorrelated. The

power spectrum due to all cosmic string wakes is the sum of the power spectrum due

to each cosmic string wake.

Figure 4.2: Space time diagram showing a wake formed at time tj crossing the past light cone at time t.

This section calculates the power spectrum due to a single cosmic string wake as a function of the time it was formed (tj) and the time it crossed the past light cone

(t). Using the scaling solution we ﬁnd the total number of such wakes. Since the diﬀerent wakes are uncorrelated, the total power spectrum is then the sum of the power spectrum from each cosmic string wake. Chapter 4: Polarization Signal Due to Cosmic String Wakes 39

4.1.1 Polarization Power Spectrum Due to a Single Cosmic

String Wake

Using equations 4.1, 4.6 and 3.8, the polarization power spectrum for a single wake

is given by:

2 ∆,∆ ∆,∆ 1 ρP 2 2 ik(cos[θ](x1 y1)+sin[θ](x2 y2)) x1 y1 CP dθ d x d ye − − (4.8) l ≈8π2 2 ∆ ∆ Z Z0,0 Z0,0 The single wake power spectrum can be written as:

2 1,1 1,1 2 1 (ρP ∆) 2 2 2 il∆(cos[θ](x1 y1)+sin[θ](x2 y2)) l CP (l∆) dθ d x d ye − − x y (4.9) l ≈2 8π2 1 1 Z Z0,0 Z0,0 1 ρ ∆ 2 P [l∆] ≡2π 4 F 2 1,1 1,1 4s 2 2 is(cos[θ](x1 y1)+sin[θ](x2 y2)) [s]= dθ d x d ye − − x y (4.10) F 2π 1 1 Z Z0,0 Z0,0 In the last step, the shape function ( [x] as shown in figure 4.3) has been defined in F such a way as to fix:

[x] lim F =1 x 0 2 → x

4.1.2 Total Power Spectrum Due to All Cosmic String Wakes

Since each cosmic string wake is expected to be statistically independent of any other cosmic string wakes, the power spectrum for all cosmic string wakes is simply the sum of the power spectrum for each cosmic string wake. Chapter 4: Polarization Signal Due to Cosmic String Wakes 40

F@sD 5

s 5 10 15 20

Figure 4.3: Shape function as deﬁned in equation 4.10 where [s] is plotted against F s.

Since cosmic strings can only accrete matter during matter domination, wakes only started growing after equal matter and radiation. As such, we need only consider times tj >teq. In order to give rise to polarization signals at late times, photons that were polarized by a cosmic string wake should not scatter oﬀ other ionized matter.

Therefore, only cosmic string wakes that cross the past light cone at times t>tls need to be considered. Since the polarization signal is proportional to the ionization fraction inside the wakes, we would expect the dominant contribution to the power spectrum to come from wakes close to the surface of last scattering or wakes that cross the past light cone after reionization. Since the magnitude of polarization is proportional to (z[t]+1)2, wakes that cross the past light cone after reionization will give rise to a much smaller contribution than wakes that cross just after last scattering.

The full polarization power spectrum due to wakes is clearly dominated by wakes that were created shortly after equal matter and radiation and that cross the past light Chapter 4: Polarization Signal Due to Cosmic String Wakes 41

cone at last scattering. If we assume that wakes remain ionized for a time δtls after

last scattering, the full polarization power spectrum can be found by multiplying the

dominant single wake solution by the number of dominant wakes as the fractional

time that wakes will remain ionized after last scattering δtlsH[tls].

Numerically, there is very little diﬀerence between the full solution and the solution

obtained by considering only wakes created at equilibrium and crossing at the last

scattering surface.

Since a wake remains at a ﬁxed co-moving distance, the number of wakes laid down

at time tj that will cross the past light cone is given by:

3 t0 dτ(z[τ]+1) tj Ntotal =nH 1 (Rz[tj]+1)H[tj]− !

The number of wakes formed at time tj that will cross the past light cone at time t

is then given by3:

2 t 1/3 t 1/3 t δ [t,t ] =12n 1 0 n j H − t t t 0 ! 0 j 3/2 1/2 12n (z[t ]+1) (z[t]+1)− (4.11) ∼ H j

Using this and equation (4.10), one can write the total power spectrum as:

3More accurately, this is the number of wakes that cross the past light cone within one Hubble time step of t. Chapter 4: Polarization Signal Due to Cosmic String Wakes 42

2 l(l + 1) P δ [t,t ]l CP [t,t ] (4.12) Cl ≈ n j l j t,t Xj ρ [t,t ]∆[t,t ] 2 δ [t,t ] p j j [l∆[t,t ]] ≈ n j π F j t,t Xj 7 2 3/2 7/2 l 1.5 10− n f(GµQ) (z[t ]+1) (z[t]+1) ≈ × H j F 2(z[t]+1)1/2 t,t Xj (4.13)

We see that the dominant contribution comes from wakes formed earliest and crossing the past light cone earliest. The earliest that a wake can form is teq. Any wake which crosses the past light cone before equal matter and radiation will cause polarization, but those polarized photons will interact will the free electrons and therefore not be observable today. The earliest photons which will still be observable today cross the past light cone shortly after the surface of last scattering, when the wakes are still ionized but the rest of the matter in the universe has already recombined. Based on these arguments, one ﬁnds that the dominant contribution to the power spectrum is given by:

2 l(l + 1) P δ [t ,t ]l CP [t ,t ] Cl ≈ n ls eq l ls eq ρ [t ,t ]∆[t ,t ] 2 δ [t ,t ] p ls eq ls eq [l∆[t ,t ]] ≈ n ls eq π F ls eq 7 δtls 2 3/2 7/2 l 1.5 10− n (GµQ) (z[t ]+1) (z[t ]+1) ≈ × H t eq ls F 2(z[t ]+1)1/2 ls eq (4.14)

Here δtls is the fraction of a Hubble time step after last scattering that wakes will tls remain ionized. Assuming that the number of wakes created per Hubble volume per Chapter 4: Polarization Signal Due to Cosmic String Wakes 43

7 Hubble time is n 10, that the cosmic string tension is Gµ 10− , that the average H ∼ ∼ CMB quadrupole moment is Q 30µK, and that the wakes remain ionized for a full ∼ Hubble time step after last scattering (f 1), the approximate upper bound for the ∼ B B power spectrum is shown in ﬁgure 4.4. −

l l+1 C BB I M l ΜK2 2 Π H L

0.025

0.020

0.015

0.010

0.005

l 500 1000 1500 2000

Figure 4.4: Estimated upper bound B B mode power spectrum due to cosmic string − wakes.

Figure 4.54, shows the B B polarization power spectrum upper bounds as well as − the predicted B B power spectrum in grey. This is made up of a lensing contribu- − tion (dominant at large multipoles) and a primordial gravitational wave contribution

(dominant at small multipoles), both of which are shown in dashed grey. Studing

ﬁgures 4.4 and 4.5 the contributions to the B B power spectrum from cosmic string − wakes has a peak at a similar position to that expected from the lensing contribu-

tion to the B B modes power spectrum, but are slightly smaller than the expected − lensing contribution.

4This ﬁgure is taken from [47]. Chapter 4: Polarization Signal Due to Cosmic String Wakes 44 ) 2

K TE BICEP CBI µ

( 100 QUaD MAXIPOL WMAP Boomerang π CAPMAP DASI / 2 l 0 l(l+1)C

-100

10 2 EE 10

-1 10

-2 10 10 2 BB: 95% confidence upper limits 10

1 -1 10 -2 10 -3 10 2 3 10 10 Multipole l

Figure 4.5: Current bounds as well as the expected B B power spectrum. −

Since the dominant contribution to the power spectrum comes from wakes formed shortly after matter-radiation equality, we expect the peak position to correspond roughly to the angular scale of the comoving Hubble radius at matter-radiation equality. This scale also determines the turnover in the matter (and therefore lensing) power spectrum. It is therefore not surprising that the spectrum due to lensing would look similar to the spectrum due to cosmic string wakes.

It should be noted that at the time this thesis was completed, there have been detec- tions of B B polarization including the primordial power spectrum due to tensors − in the early universe [7] as well the lensing B B signal by both the South Pole − Telescope (SPT) and POLAR-BEAR[86, 6]. Chapter 4: Polarization Signal Due to Cosmic String Wakes 45

4.2 Conclusions

Cosmic string wakes give rise to a particular position space polarization signal as

studied in [59]. This study aimed to ﬁnd the corresponding signal in the polarization

power spectrum. This study ﬁnds that the contribution to E E and B B mode − − power spectra is the same. For values of the string tension satisfying the current

7 bound (Gµ 10− ), the contribution to the E E mode power spectrum from ∼ − cosmic strings is much smaller than the background as predicted by concordance cosmology. The correction to the B B power spectrum due to cosmic string wakes − is smaller than the B B power spectrum induced by lensing. − Since the shape and size of the induced B B mode power spectrum due to lensing − is similar to the shape of the B B mode polarization due to cosmic stings, it will be − diﬃcult to distinguish these two signals, unless two peaks can be distinguished near l 1000. It seems unlikely that searching the polarization power spectrum will lead ∼ to any improved bounds on cosmic string wakes.

We can conclude that searches for cosmic string wakes in polarization maps will be a great deal more successful in position space than in Fourier space. This is due to the distinct pattern that arises from the cosmic string wakes. In particular, using an edge detection algorithms such as the Canny algorithm [45] in B-mode polarization maps give the best hope of improving bounds on the string tension using CMB polarization. Part II

Redshift Luminosity Relation,

Curvature and Dark Energy

46 Chapter 5

Introduction: Luminosity Distance

Redshift Relation

This chapter aims to introduce the luminosity distance redshift relation, as well as

its signiﬁcance in modern cosmology.

5.1 FLRW Universes

An FLRW universe is an n + 1 dimensional pseudo-Riemannian space with an n

dimensional maximally symmetric subspace. In 3+1 dimensions, an FLRW universe

can be written with a metric:

2 2 2 2 dr 2 2 2 2 ds = a[η] dη + κr2 + r (dθ + sin[θ] dψ ) (5.1) − 1 ! − 6 κ Here a[η]2 is the Ricci scalar on the subspace. A universe with κ < 0 is said to be open, a universe with κ > 0 is said to be closed, and a universe with κ = 0 is said

47 Chapter 5: Introduction: Luminosity Distance Redshift Relation 48

to be spatially ﬂat. a[η] is called the scale factor. The quantities a[η] and κ are not observables. They are only deﬁned up to some normalization.

This universe must obey the Einstein equations:

1 8πG R Rg = T (5.2) µν − 2 µν c4 µν

Tµν is the stress energy tensor and is described in section 5.1.1. In what follows, units

are chosen such that 8πG = 1 and c = 1.

5.1.1 Stress Energy Tensor

The most general form for the stress energy tensor of a ﬂuid is given by:

1 T =(ρ + P ) U U + Pg + (U Q + U Q ) + Π µν µ ν µν 2 µ ν ν µ µν

µ where 0=Q Uµ

µν 0 =Π Uµ

µ 0 =Πµ

Here, ρ is the energy density of the universe, P is the pressure of the universe, Uµ is

the 4-velocity of set of observers measuring the ﬂuid (one going through each event

µ µ in spacetime), Q is the heat ﬂux of the ﬂuid relative to the observers, and Πν is

the anisotropic stress as measured by the observers. This parameterization is not

invariant, as diﬀerent observers will see diﬀerent values for these parameters. For a

set of observers comoving with the ﬂuid, Qµ = 0. We also assume that there is no Chapter 5: Introduction: Luminosity Distance Redshift Relation 49

anisotropic stress for the ﬂuid when it is measured by a comoving observer (Πµν = 0).

Therefore, if we consider observers who are comoving with the ﬂuid, the stress energy tensor is given by:

Tµν =(ρ + P ) UµUν + Pgµν (5.3)

In the concordance model, we usually consider five separate fluids: photons, neutrinos, baryonic matter, nonbaryonic dark matter, and dark energy. These different species interact in a nontrivial way. In what follows we will approximate photons and neutrinos as a single perfect fluid (which we will call radiation hereafter) with an equation of state (ω P = 1 ). Since radiation and baryonic matter have decoupled ≡ ρ 3 at late times, matter and dark matter can be treated as a single perfect fluid with an equation of state (ω 0). Dark energy is treated as a cosmological constant, ≈ which will only be included perturbatively at second order. Since this study aims to deal with late times, the radiation density will not significantly affect the geometry.

Ultimately, results will be quoted up to zeroth order in the radiation density.

5.1.2 Friedmann Equations

For the metric given in equation (5.1), the Einstein equations become:

a [η]2 ρ[η] κ ′ = a[η]4 3 − a[η]2 a [η] 1 κ ′′ = (ρ[η] 3P [η]) a[η]3 6 − − a[η]2

1 a′[η] ρ′[η]=− (ρ[η]+ P [η]) 3 a[η] Chapter 5: Introduction: Luminosity Distance Redshift Relation 50

Here a prime denotes a derivative with respect to the conformal time parameter, η.

The third equation is called the continuity equation and is not independent from the

first two equations. In the case that we have a perfect fluid, we find that:

ρ [η] 1+ ω a [η] ′ = ′ ρ[η] − 3 a[η]

This leads to:

3(1+ω) ρ a[η]− ∝

5.2 Standard Candles and Type IA Supernova

Light, emitted from a particular event, travels, without interacting, through an ex-

panding universe until it is observed, at another event. As the light travels through

spacetime its wavelength is also stretched by the same amount that spacetime has

δλ stretched since it was emitted (z = λ ). Therefore, if we know both how long ago a photon was emitted, as well as the wavelength of that photon when it was emitted, then we have an expression for how much spacetime has expanded between the emission and the observation of the photon. In principle, given this information, we can accurately solve for the geometry of the universe (at least since photons decoupled from matter). Finding photons with a well-known frequency emitted at a well-known distance is not simple. Sources that have a well-known photon emission spectrum and are at a well-known distance are diﬃcult to ﬁnd, and are known as standard candles. Chapter 5: Introduction: Luminosity Distance Redshift Relation 51

The most famous standard candle is the surface of last scattering. Since the elec-

tromagnetic interaction is well understood, the temperature at which photons and

electrons decouple is known. We can also measure the current temperature of the

radiation ﬁeld, allowing an accurate calculation of the redshift since last scattering.

The ﬁrst acoustic peak of the CMB temperature anisotropy power spectrum can be

used as a standard ruler, giving an accurate measurement of the distance to the last

scattering surface. We therefore know how much spacetime has expanded since de-

coupling, giving us a very accurate data point in the expansion history of the universe.

Other standard candles are trickier to ﬁnd. Although the distance to nearby galaxies

can be found, galaxies come in a number of diﬀerent shapes, sizes, densities, and com-

positions. As such, they are not good standard candles. Diﬀerent chemical processes

have well deﬁned emission spectra, but computing how long ago the process occurred

is more complicated.

In searching for a standard candle, we should search for a physical process that only

occurs under very speciﬁc conditions with a well-deﬁned spectrum. The most common

example is the process of a star going supernova. Unfortunately not all stars do go

supernova, limiting the number of data points. Also, since stars only started forming

after reionization, supernova events will only happen for small redshifts (the oldest

observed supernova were found at z 2). Another more important issue is that not ∼ all supernova events happen in the same way.

Most supernova form as a result of the gravitational interaction overcoming electron degeneracy pressure, resulting in the star’s core collapsing. Type Ia supernovae are thought to result from stars with a mass just below the Chandrasekhar limit (see Chapter 5: Introduction: Luminosity Distance Redshift Relation 52

section 5.2 of [87]). Since these stars exist just below the Chandrasekhar limit, the

star does not go supernova due to core collapse, resulting in an extended period of

time during which most of the lighter elements are completely burned up due to

fusion. At some point, a thermal runaway caused by Carbon fusion will cause the

star to explode very violently. As such, these type Ia supernova have very consistent

properties. While there is a large scatter in the intrinsic brightness of the supernova

event, the maximum brightness is correlated with the decay in B magnitude[124]. As

such using the peak brightness and the decay rate allows us to accurately calculate

the luminosity distance[85], making type Ia supernovae ideal candidates for standard

candles.

5.3 Luminosity Redshift Relation

As stated in section 5.2, if we have a standard candle, we can ﬁnd an expression for

how much space has expanded between the source and the observer. Since distance is

not as well deﬁned as we might hope, we pick a particular deﬁnition of distance with

which to work. The most standard deﬁnition, keeping the idea of a standard candle,

in mind is the expression for the luminosity distance (dL) deﬁned by:

L d2 = (5.4) L 4πF

Here, L is the absolute bolometric luminosity of the standard candle, and F is the bolometric ﬂux due to the candle as measured by the observer. Since the absolute luminosity of all supernovae can be found, the luminosity distance can be measured Chapter 5: Introduction: Luminosity Distance Redshift Relation 53

directly from the bolometric ﬂux of supernovae. It is also worth deﬁning the area

distance (dA) and the distance modulus (µ):

2 dA =(1 + z) dL d µ =5log L (5.5) 10 d 0

The constant d0 attached to the deﬁnition of the distance modulus is needed to ensure

5 the expression remains dimensionless. One usually takes d0 = 10− Mpc.

Equipped with a well-deﬁned expression for distance and redshift, we can set about

ﬁnding an expression relating the luminosity distance to the redshift for various sources, thereby solving for the geometry of spacetime.

The expression for the luminosity distance as a function of redshift is rather simple in a perfect FLRW universe. In a ﬂat space time, the ﬂux must be proportional to

2 d− , where d is the physical distance between the source and the observer. In an expanding spacetime, the photons will be redshifted, and therefore the ﬂux will be

2 decreased by a factor (1 + z)− . The physical distance between the source and the

observer at the time of observation (η0) will then be given by a[η0]r, where r is the

coordinate distance deﬁned by the metric given in equation (5.1). Using the deﬁnition

of the ﬂux in terms of the luminosity distance we ﬁnd that:

dL =r[z]a[ηo](1 + z) (5.6)

In the case of a universe containing perfect ﬂuids, this expression becomes: Chapter 5: Introduction: Luminosity Distance Redshift Relation 54

z dζ dL =(1 + z)χκ 3 2 4 0 H Ω + Ω (1 + z) + Ω (1 + z) + Ω (1 + z) "Z 0 Λ m κ r # p (5.7)

sin[x] κ> 0  where χκ[x]=  x κ =0 (5.8)   sinh[x] κ< 0   Here χκ is a function which is dependent on the spatial curvature of spacetime. Figure 5.1 shows the luminosity distance redshift relation in a spatially flat FLRW universe, containing a single perfect fluid with equation of state ω = 1, 0, 1 . {− } Figure 5.1 shows an open, flat and closed FLRW universe containing matter alone.

dL Ho 6

4 Ω=-1 Ω=0 3 Ω=1 2

0.5 1.0 1.5 2.0 z

Figure 5.1: Figure showing the luminosity distance as a function of redshift for a ﬂat

FLRW universe containing a single perfect ﬂuid with equation of state ω = 1, 0, 1 { − } plotted in “dotted blue”, “solid green” and “dashed red” respectively.

In general, the luminosity distance redshift relation has dimensions. In ﬁgures 5.1 and 5.2, the slope of the graph is inversely proportional to the square root of the total Chapter 5: Introduction: Luminosity Distance Redshift Relation 55

dLH0

2.5

2.0 W= 3 Κ>0 2 H L 1.5 Out[142]= W=1HΚ=0L W= 3 Κ<0 1.0 4 H L

0.5

z 0.5 1.0 1.5 2.0

Figure 5.2: Figure showing the luminosity distance as a function of redshift for an

FLRW universe containing matter with positive zero and negative spatial curvature plotted in “dotted blue”, “solid green” and “dashed red” respectively. energy density plus the spatial curvature of the universe. The shape is dependent on the type of energy (or curvature) being considered. We can see that either lowering the equation of state, decreasing the spatial curvature, or decreasing the total energy content of the universe leads to a decreased redshift (or less expansion) for objects at the same luminosity distance. Conversely, raising the equation of state, increasing the spatial curvature, or increasing the total energy content of the universe leads to increased redshift (or more expansion) for objects at the same luminosity distance.

5.4 Luminosity Redshift Relation in a General Space-

time

We now turn our attention to the expression for the luminosity distance. An expression for the area distance in an FLRW universe was given in equation 5.6. Since the universe is not perfectly FLRW, it makes sense to study the eﬀect of inhomogeneities Chapter 5: Introduction: Luminosity Distance Redshift Relation 56

and anisotropies on the expression for the luminosity distance.

Since the universe is no longer considered to be homogeneous or isotropic, measuring

the luminosity redshift relation will be dependent on both the position of the source

and the observer.

Consider, for example, an observer centered on an over density measuring objects at

distances much further away than the extent of the over density. Such an observer

will see a reduced redshift in his measurements, since being in a potential well relative

to the average will result in him experiencing a small additional blueshift. Similarly,

a source centered in an under density will also result in a decreased redshift. This

will also result in the luminosity distance redshift relation acquiring a directional

dependence.

In order to fully specify the luminosity redshift relation in a general spacetime, we

µ must fully specify the position of an observer (xobs). Once this is speciﬁed, a particular

ν source must be chosen (xsource). Since photons are being observed, any source should

lie on the past light cone of the observer. Each source can then be labeled by two

angles (θa) which specify the direction of the source from the observer. Finally, the

depth of the source from the observer can be labeled by the amount by which light

has been redshifted between the source and the observer (z). In the most general spacetime, we can write the luminosity distance as:

d [xµ ,xν ] d [xµ ,θa,z] (5.9) L obs source 7→ L obs

This is in contrast to the same expression in a perfect FLRW universe where: Chapter 5: Introduction: Luminosity Distance Redshift Relation 57

dFLRW [xµ ,xν ] dFLRW [η ,z] (5.10) L obs source 7→ L obs

In order to reconcile the eﬀects of structure, it makes sense to average over sources

and observers in such a way as to be able to compare d [xµ ,θa,z] [η ,z] to h L obs i obs FLRW dL [ηobs,z].

5.5 Dark Energy: How Exactly is it Constrained

by Observations?

In the concordance model, dark energy is needed to explain a number of observations. In particular, SN1a data, measurements of the anisotropies in the cosmic microwave background (both temperature and polarization), large scale structure

(including baryon acoustic oscillations), and gravitational lensing are important measurements in cosmology.

In each case, we need to establish what exactly is being determined. Supernova data allows us to ﬁnd an expression for the luminosity distance dL[z] for small redshifts

(z < 2).

CMB data allows for an accurate calculation of the area distance (and therefore of the luminosity distance) to the surface of last scattering (dL[zls]). Aside from the integrated Sachs Wolfe eﬀect, the CMB power spectrum does not directly constrain the cosmological constant (Λ). It does, however, constrain the physical baryon and

2 2 matter densities (Ωbh and Ωmh ) as well as the scalar spectral index (ns) [150]. Chapter 5: Introduction: Luminosity Distance Redshift Relation 58

The Sloan Digital Sky Survey (SDSS) has mapped out galaxies for z . 0.3[9], allowing for an accurate measurement of the galaxy power spectrum. This measurement gives the power spectrum as a function of z, which can in principle determine all the cosmological parameters. Since we do not fully understand the nonlinear evolution of the power spectrum on small scales, or the correlation between galaxies and matter on large scales [19], we are limited in the conclusions that can be drawn. Finally, since the survey has only mapped galaxies out to z 0.3, the evolution of the power ∼ spectrum is diﬃcult to constrain.

While these experiments do constrain Λ independently of dL[z], these bounds are not as tight as might be expected from quoted results. In particular, the independent constraint on Λ from the CMB comes from the late time integrated Sachs-Wolfe effect, which in turn comes from the evolution of the scalar fluctuation. Similarly the independent constraint on Λ from galaxy surveys comes from the evolution of the power spectrum, which again comes from the evolution of the scalar fluctuation.

The degeneracy between Λ and dL[z] is broken if we can ﬁnd an expression for the late time evolution of ﬂuctuations in the universe. Since this evolution is nonlinear at late times, it is not as well understood. As such, the constraints on Λ that are independent of dL[z] are not as well understood.

5.6 Overview of the Problem

This part of the thesis has been organized as follows: This chapter (chapter 5) has introduced the luminosity distance redshift relation, and its central role in modern cosmology. It has also brieﬂy mentioned the dependence of the luminosity distance Chapter 5: Introduction: Luminosity Distance Redshift Relation 59 redshift relation on the details of the source and the observer, motivating the need to average over both the source and the observer. There will be a brief discussion of some previous work in which this problem has been studied.

In chapter 6, we set up the important features of studying this problem. In section 6.1 we describe the geometry of the universe we will study. A number of expressions, which will be of importance when it comes to averaging in chapter 10, are then described. The evolution of a subdominant ﬂuid (radiation) during matter domination

(section 6.3) is described. In section 6.4, geodesics and quantities that are conserved along the past light cone are also studied and related to the coordinates used in the so called geodesic light cone gauge (GLC [76]). We then study the power spectrum as well as a number of approximations which will be of use in chapter 10. A general expression for the area distance is given in chapter 7.

Chapter 8 begins with a discussion on geometric and ensemble averages. Embedded submanifolds and the average of embedded submanifolds are then discussed leading to a general expression for geometric averaging of an observable over a submanifold.

We then discuss spatial averaging and the role of a clock, followed by light cone averaging, and the importance of redshift. An expression for the combined light cone and spatial averaging is then given. The process of evaluating averaged expressions up to second order in perturbation theory is then described.

Chapters 9 and 10 uses the formalism built in chapter 8. Chapter 9 deals with averaging observables which can be deﬁned locally (in particular they do not depend on a source, but only on an observer). Expressions for the average matter density, spatial curvature and cosmological constant, as well as the averaged expansion parameter Chapter 5: Introduction: Luminosity Distance Redshift Relation 60

and deceleration parameter, are found. There is also a discussion on long wavelength

ﬂuctuations and conditions under which they emerge in global observables. Chap-

ter 10 then ﬁnds an expression for the luminosity distance redshift relation in terms

of the averaged matter density, spatial curvature and cosmological constant. We then

apply this in a number of limits, in particular we ﬁnd the relationship between the

slope of the luminosity distance redshift relation and the expansion as calculated in

chapter 9.

Finally in chapter 11, the expression for the averaged luminosity distance redshift

relation obtained in chapter 10 is then ﬁtted to the data from [137], and [150] in

order to estimate the averaged matter content, the spatial curvature and cosmological

constant of the universe.

5.7 Averaging in Gravity and Previous Work Re-

lating to Backreaction and Expansion

Since gravity is a nonlinear theory, the question of averaging gravity is nontrivial.

Almost independently of what exactly is meant by averaging, we will ﬁnd that:

G [ g ] = G [g ] µν h αβi 6 h µν αβ i

Here indicates some form of averaging, which is as yet undeﬁned. Since the ΛCDM hi model is based on the assumption that the universe is homogeneous and isotropic, the most common starting point for any cosmological model is to begin with the ansatz that the average of the metric is an FLRW metric (a four dimensional metric with a Chapter 5: Introduction: Luminosity Distance Redshift Relation 61

maximally symmetric three dimensional submetric). A common starting point is to

consider:

G [ g ]= T [ g ] (5.11) µν h αβi µν h αβi

Clearly, a more reasonable approach would be to consider:

G [g ] = T [g ] (5.12) h µν αβ i h µν αβ i

The main reason that equation (5.11) is used more often than equation (5.12) is due

to the complexity of the Einstein tensor, in that G [ g ] is far easier to work with µν h αβi than G [g ] . We can approximate the write the full metric as: h µν αβ i

g = g + δg (5.13) αβ h αβi αβ

If we were to consider corrections up to linear order in δgαβ, we would have:

δ G [g ] G [ g ]+ δg G [ g ] (5.14) h µν αβ i≈ µν h αβi γλ δ g µν h αβi h γλi If perturbation theory is only treated up to linear order, then δg = 0. In this case h αβi we must ﬁt an FLRW universe to the cosmological data. This is the usual approach.

A less popular opinion is that the nonlinear corrections (usually called backreaction) are important, and could have significant effects. Since the cosmological data fits the

FLRW background so well, and corrections to the metric are so small, we might argue that the eﬀect of second order corrections must be small. Chapter 5: Introduction: Luminosity Distance Redshift Relation 62

At this point, there are two important questions to ask, namely, “What exactly is

meant by averaging?” and “How big can this eﬀect be?”.

Averaging is diﬃcult at a conceptual level. There are a number of approaches to the

averaging problem, each of which comes with certain technical or practical problems.

The ﬁrst approach might be to consider averaging the Einstein equations. Since

gravity is geometric and the spacetime is not in general ﬂat, vectors and tensors

(such as the Einstein equation itself) are not invariant under rotations or translations.

Since vectors and tensors at diﬀerent positions cannot be added in an obvious way,

and since averaging implicitly requires the notion of addition, deﬁning averaging for

vectors and tensors is diﬃcult1.

Averaging over scalar quantities, however, is easier to deﬁne [52]. Perhaps the most famous formalism for spatial averaging is the one suggested by Buchert [41, 39, 40, 38].

Extending the ideas presented by Buchert, Carfora and Piotrkowska[46] use a renormalization group approach to try and understand how the scale over which averaging is performed can affect the averaged effects on cosmological parameters. More recently, averaging over scalar quantities has been discussed by Gasperini and collaborators [77, 78, 76]. See also works by [36]. While this approach seems promising, in order to fully account for a general space time, one would need to fully define ten

1See works by Zalaletdinov[53, 159, 160], who tries to deﬁne averaging by making use of bitensors, which are not uniquely determined. Chapter 5: Introduction: Luminosity Distance Redshift Relation 63

independent physical scalars2.

Another approach is to consider an eﬀective energy momentum tensor, ﬁrst discussed by Issacson in [93, 94] and suggested in a cosmological setting in [1].

In order to answer the question “How big can this eﬀect be?”, we must be more speciﬁc. Since most cosmological observables (take an arbitrary observable for O example) are nonlinear in gµν, a second order perturbative treatment will give rise to

at least some small corrections. Since is an observable, it must be gauge invariant. O In order to get an estimate of the importance of nonlinear eﬀects, we might consider

the dimensionless quantity:

[g ] hO αβ i 1 (5.15) [ gαβ ] − O h i 2 ∂ [ gστ ] 1 δg δg O h i ( [ g ])− + ... (5.16) ≈h αβ λγi ∂ g ∂ g O h κζ i h αβi h λσi If the magnitude of this quantity is small ( 1), then the correction is small and ≪ backreaction is unimportant. If the magnitude of this quantity is large (& 1), then

the corrections are large, and backreaction is important. In particular, if this object

is much larger than unity, perturbation theory breaks down, and nonperturbative

techniques will be needed to accurately ﬁnd the size of the eﬀect (alternatively, higher

order terms need to be added to the perturbative expansion making the calculation

increasingly diﬃcult). If all physical observables have small amounts of backreaction,

2In the case that we consider a linearly perturbed cosmological background, we have four scalar, two vector, and two tensor degrees of freedom. The assumption that only scalar perturbations are relevant makes a choice about the two vector and two tensor degrees of freedom, which would otherwise have to be mapped to some well-defined scalars. Of the four scalar degrees of freedom, two are mapped into gauge choices or, stated alternatively, how we choose to label points on the manifold. Of the last two, one is often absorbed into the assumption that there is no anisotropic strain, leaving a single scalar degree of freedom at first order. Chapter 5: Introduction: Luminosity Distance Redshift Relation 64 then the approximation in equation (5.14) is justified. If, however, some physical observables have large backreaction effects, we must challenge the approximation made in equation (5.16).

The study of backreaction and the size of the eﬀect has been studied by a number of authors. In what follows we describe a rough timeline of the discussion of the importance of backreaction:

Woodard and Tsamis [144, 145] ﬁrst argued that IR modes in gravity relax the cosmological constant in a de-Sitter spacetime. Since then these two authors have written a number of papers studying the quantum backreaction eﬀects in gravity.

This study was picked up by Mukhanov, Abramo, and Brandenberger [113, 1] where the authors studied the backreaction effects of scalar perturbations. They argued that in the long wavelength limit, that these fluctuations will relax the cosmological constant as was found by Woodard and Tsamis. These studies lead Unruh [147] to argue that long wavelength fluctuations locally look like gauge transformations, and that locally one cannot detect the effects of IR modes, and therefore cannot lead to a relaxing of the cosmological constant. This result was confirmed by Geshnizjani and

Brandenberger in [80]. The same authors then argued in [81] that if one considers entropy modes, that IR fluctuations could lead to observable consequences. This was confirmed in an inflationary context by Marozzi, Vacca, and Brandenberger[108].

Since entropy fields were required in order to get corrections which were sensitive to long wavelength fluctuations, Kolb, Matarrese, Notari, and Riotto [104] calculated the effect of inhomogeneities on the expansion rate of the universe. They found that the size of these effects must be small. This lead Barausse, Matarrese, and Riotto [15] Chapter 5: Introduction: Luminosity Distance Redshift Relation 65 to study the effects of inhomogeneities on the luminosity distance redshift relation, where they argue that the effect of inhomogeneities is small, but could play a part in explaining the size of the present day acceleration of the universe. Geshnizjani,

Chung, and Afshordi[82] argued that the corrections found by [15] could not explain present day acceleration but only the correction to the size of spatial curvature terms.

In particular they argue that in the context of inﬂationary models, this eﬀect is very small.

This study has been picked up by many author since then. In particular it has been studied in the context of local observables by a number of authors.

Among these authors are Nambu and collaborators [115, 96] place LTB patches in a “Swiss cheese” model of the universe, where they find that the cosmic volume expansion is increased by these inhomogeneities. They also argue that this effect is due to non linearities which might only appear beyond third order in perturbation theory. Wiltshire has studied the effects of backreaction in a number of papers, where he argues that our position in spacetime affects our measurement. Wiltshire makes use of a two scale “timescape model”[153, 155, 154, 156, 157] where the universe is split up into a fraction of voids and walls. He argues that with such a model, there is no need for dark energy. Kolb, Matarrese and other collaborators [104, 102, 109, 110, 101, 100] have used a “Swiss cheese” to study the possibility that acceleration can be caused by inhomogeneities. Chung and collaborators [48] have studied the effect of backreaction in an LTB model, where the effective curvature dominates. While Buchert has mainly focused on the role of averaging, he and various collaborators[107, 152, 43, 42] have studied the effects of backreaction and its relation to structure formation. In these Chapter 5: Introduction: Luminosity Distance Redshift Relation 66

works, they ﬁnd dynamic corrections to the dark energy, dark matter, and spatial

curvature of the universe.

Many consider the eﬀects of backreaction to be small. Consider the argument by

Ishibashi and Wald [95]. Consider a universe described by a perfect FLRW universe

with energy density given by ρ = ρ0 + δρ. In the linear regime, we can assume that

inhomogeneities are nearly described by the Newtonian potential Φ with Φ 1 and | | ≪ obeying the Poisson equation 2Φ=4πGδρ. In this case, we find that ∂ Φ is small ∇i t and as such cannot significantly affect the evolution of the universe. Since the effects after averaging can only arise at second order, the first nonvanishing correction to the expansion is Φ2, and, as such, must remain small, in spite of the potential for δρ ∼ ρ0 to be large. In the same paper these authors argue that one could build a universe with an average accelerated expansion while not allowing for any local region in which space is accelerating. As such, averaged local acceleration is not a useful observable.

In a recent paper Green and Wald [84] argue that large matter fluctuations on small scales cannot significantly affect the dynamics of the concordance model. However, in response Korzynski [106] has constructed an explicit exact solution with a nested hierarchical structure of overdensities and voids in which the specific mathematical assumptions of Green and Wald do not apply, and in which the effects of backreaction are significant - a result which can be attributed to the nonlinearity of adding masses in general relativity.

Paranjape and Singh [117, 120, 118, 119] who have discussed averaging and the pos- sibilities that inhomogeneities can lead to accelerated expansion. They find that the effects are too small to significantly affect cosmic dynamics. Räsänen argues that the Chapter 5: Introduction: Luminosity Distance Redshift Relation 67 effects of backreaction can largely be accounted for as shifts in the spatial curvature and matter density in [132]. He has also discussed accelerated expansion in relation to structure formation [131], as well as other works related to linear perturbation theory and LTB models [130, 129, 128]. He argues that backreaction cannot have significant effects. Durrer and collaborators [2, 22] argue that small scale structure cannot have significant effects on cosmological evolution.

Instead of studying local expansion rates, some authors have studied the luminosity distance redshift relation itself. In doing so they can bypass the arguments by

Ishibashi and Wald which relate to the local expansion rate of the universe.

As already mentioned Barausse and collaborators[15] studied corrections to the luminosity distance redshift relation, and then performed the small redshift expansion of their result.

In each other case of studying the eﬀects of inhomogeneities on the luminosity distance redshift relation, various authors identify two important corrections, namely redshift space distortions at small redshift, and a lensing eﬀect at large redshifts. Bonvin,

Durrer, and Gasparini [33] calculate the power spectrum of first order fluctuations in the luminosity distance redshift relation. Pyne and Birkinshaw[127] study the first order correction to the luminosity distance redshift relation, as well as the lensing correction to the cosmic variance. In [92] Hui and Greene study the effects of peculiar motion on the luminosity distance redshift relation.

Recently, the work by Ben-Dayan, Gasperini, Marozzi, Nugier, and Veneziano regarding light cone averaging [20, 21] has been used to study expressions for the luminosity redshift relation, as well as its variance rather than the expansion rate. They argue Chapter 5: Introduction: Luminosity Distance Redshift Relation 68

that inhomogeneities cannot account for dark energy. In [22] the authors argue for

added uncertainty in measurements of the local Hubble rate based on the works by

Gasperini et al. They ﬁnd that this additional uncertainty relaxes the conﬂict be-

tween the value of H0 obtained from CMB data and supernova data. Clarkson, Umeh,

Maartens, and Durrer [50, 146] have recently found corrections to the area distance redshift relation, and argue that the small corrections to the area distance to the surface of last scattering can lead to corrections to the values of various cosmological parameters of the order of 1σ

In this thesis, we do not address the question of cosmic evolution by studying the expansion3. Instead we choose to study an observable that is used as a trace of cosmic

evolution, namely the area distance. This has been considered recently by Gasperini

and collaborators [20, 21, 58]. In their work they pay particular attention to the role

of the source.

In this work we extend the calculation of Ben-Dayan and collaborators, paying at-

tention to the role of both the source and the observer. We ﬁnd observer dependent

corrections to the luminosity distance can mostly be accounted for by a shift in the

average spatial curvature and energy density, which is consistent with the ﬁndings

of [132]. We also note that corrections that are sensitive to long wavelength ﬂuctua-

tions only contribute to a shift in the average matter density. This is in contrast to

the ﬁndings of [113, 1]. Corrections that cannot be accounted for by a shift in the

average matter density or average spatial curvature are dominated at small redshifts

3However, the relationship between the slope of the luminosity distance redshift relation and the expansion rate of the universe is tested. The expression for the deceleration parameter is also studied. Chapter 5: Introduction: Luminosity Distance Redshift Relation 69 by redshift space distortions, and at large redshifts by a lensing correction, which are consistent with [20, 21, 33, 92, 127]. Our work shows that the usual small redshift expansion of the luminosity distance redshift relation is not related to the expansion and the deceleration parameters as it is in a perfectly FLRW universe, and therefore we argue the approach of [15] is not valid. Chapter 6

Perturbing an FLRW Universe

We begin with the assumption that the universe is nearly FLRW with small perturbations. We also make the assumption that the universe is statistically homogeneous and isotropic. This statement will be discussed in section 6.1.2, but at the most basic level, we presume that the metric after ensemble averaging must be an FLRW metric.

We do not assume ergodicity [75] in this work, although we ﬁnd that ergodicity holds at ﬁrst order. In what follows, a universe containing matter, as well as negligible amounts of noninteracting radiation, is considered. At second order, dark energy as well as curvature are included.

Once the geometry is built, various important objects are studied, which ultimately lead to an expression for the luminosity distance.

70 Chapter 6: Perturbing an FLRW Universe 71

6.1 Solving for the Geometry

In what follows, we start with a matter dominated FLRW universe at zeroth order. At

ﬁrst order, we consider scalar matter perturbations. At second order, we include the

minimal number of terms needed to ensure that the Einstein equations are obeyed

after ensemble averaging. This includes homogeneous, isotropic corrections to the

cosmological constant, matter density and curvature.

6.1.1 Zeroth Order Flat FLRW Matter Dominated Universe

As mentioned in chapter 5, a spatially ﬂat FLRW universe can be written with a

metric:

ds2 = a[η]2 dη2 + dx2 + dy2 + dz2 − This must obey the Einstein equations:

1 R Rg = T µν − 2 µν µν

Where, we deﬁne:

ρ(0) T = mat U U µν a[η]3 matµ matν 1 U = 1, 0, 0, 0 matµ a[η] { }

The general solution is given by: Chapter 6: Perturbing an FLRW Universe 72

ρ(0) a[η]= mat (η η )2 12 − 0 η2ρ(0) a[η]= mat (6.1) 12

Here, η0 corresponds to an arbitrary shift in the deﬁnition of conformal time. For

convenience, we choose η0 = 0.

6.1.2 First Order Cosmological Perturbation Theory

First order cosmological perturbation theory was studied in [35]. Here we do not consider anything new. One can write the metric as:

(0) gµν =gµν + δgµν

1+2Φ ∂iB + Si 2 − − =a[η]   ∂iB + Si 2(ψδij ∂i∂jE) ∂iFj ∂jFi hij  − − − − −    Here, four scalar (Φ,ψ,E and B), four divergence free vector (Si and Fi), and two

trace free divergence free tensor degrees of freedom (hij) have been deﬁned, where

each is a function of the coordinates (~x and η). Of these, two scalar and two vector

degrees of freedom are gauge artifacts. In this work, longitudinal gauge (E = B = 0)

will be chosen for convenience. Since scalar ﬂuctuations will generally be dominant

over vector and tensor ﬂuctuations, the latter will be neglected (S = F 0,h 0) i i ≈ ij ≈ in this study. Finally, it is assumed that there is no anisotropic stress (Φ = ψ).

Under these assumptions, the ﬁrst order metric can written as: Chapter 6: Perturbing an FLRW Universe 73

ds2 = a[η]2 ( 1+2Φ[η, ~x]) dη2 +(1+2Φ[η, ~x]) δ dxidxj + ... (6.2) − ij Since the fluid contains matter moving at speeds less than the speed of light, one must enforce gµν U µ U ν = 1. Up to vector quantities, the most general expression for mat mat mat − the fluid 4-velocity at first order is then given by:

1 U µ = 1+Φ[η, ~x],∂ u [η, ~x] (6.3) mat a[η] { j mat }

µ Here umat[η, ~x] is included to allow for the most general form of Umat which obeys

g U µ U ν = 1, and will be determined using the Einstein equations. µν mat mat −

µν µ ν T =ρmatUmatUmat ρ(0) where ρ = mat (1 + δ [η, ~x]) mat a[η]3 mat

We have introduced δmat[η, ~x], which is the correction to the background matter

content, allowing for spatial dependence of the matter content of the universe.

Solving the Einstein equations for these expressions, we ﬁnd that:

Φ [~x] Φ[η, ~x] =Φ [~x]+ decay 0 a[η]5/2

We can clearly see that at late times, only Φ [~x] remains relevant. Setting Φ [~x] 0 decay ≈ 0, we ﬁnd that: Chapter 6: Perturbing an FLRW Universe 74

1 δ [η, ~x] =2Φ [~x] η2 2Φ [~x] mat 0 − 6 ∇i 0 η u [η, ~x]= Φ [~x] mat 3 0 It is worth noting here that during matter domination, the metric perturbations are

ﬁxed in time, and that the shape of the power spectrum is not determined.

Stochastic Perturbations, and Ensemble Averaging

At this point, the form of the fluctuations needs to be considered. Since we are only considering first order fluctuations with zero anisotropic stress. Without any loss of generality, we can write the metric fluctuation in terms of its Fourier transform1:

3 d k i~k ~x Φ [~x]= e · φ [k]E[~k] (6.4) 0 (2π)3/2 0 Z We have introduced the variable (E[~k]). One usually makes the approximation that

the ﬂuctuations are Gaussian, in which case E[~k] will be choosen to obey the following

properties:

E[~k] =0

~ ~ 3 ~ ~ E[k]E[k′]=δ (k + k′) (6.5)

Here we have introduced the concept of an ensemble average, which we represent

with an overline2. Each realization of E[~k] corresponds to a possible realization of

1 Notice that the metric perturbation is given by Φ0[~x] which has a Fourier transform given by ~ φ0[k]E[k]. 2Ensemble averaging will be discussed further in section 8.1. Chapter 6: Perturbing an FLRW Universe 75

the geometry of the universe. Since we do not have a detailed description of the

universe, it makes sense to consider the average of all possible realizations of the

universe. Ensemble averaging is this average over realizations of the geometry of the

universe.

The power spectrum of metric ﬂuctuations is then given by:

3 3 d kd k′ i~x(~k+k~′) Φ [~x]Φ [~x]= e φ [k]φ [k′]E(~k)E(k~ ) 0 0 (2π)3 0 0 ′ Z d3k = φ [k] 2 (2π)3 | 0 | Z k2dk = φ[k] 2 2π2 | | Z dk P [k] (6.6) ≡ k φ Z It is worth noting that Φ[~x]2 does not have any spatial dependence. Therefore, after taking the ensemble average of two ﬁrst order quantities, all that remains is a time dependent quantity. It is clear from this statement that after ensemble averaging any physical observable will be invariant under spatial rotations and translations.

Statistically, this spacetime still obeys the cosmological principle. The correlation between any two space time points is only dependent on the coordinate distance between those two points 3.

Since two ﬁrst order perturbations will give a second order term, which does not vanish after ensemble averages, we must consider second order corrections to the metric.

3This statement does not hold true in a universe with an evolving metric perturbation Chapter 6: Perturbing an FLRW Universe 76

6.1.3 Second Order Cosmological Perturbation Theory

Since, after ensemble averaging, the leading correction to any physical observable

is second order, consistency requires that the Einstein equations be obeyed up to

second order. Second order cosmological perturbation theory has been studied by a

number of authors (see [37, 114] for two examples). The analysis that follows does

not come directly from previous research but is closely related to the construction of

an “eﬀective stress energy tensor” as a tool to study backreaction (see [111] for an

example, as well as [95] for criticisms).

Since any observable will only be studied after ensemble averaging, at second order

we include only terms which survive ensemble averaging, and are homogeneous and

isotropic. We also consider corrections to the curvature, the matter content, and the

dark energy content of the universe at the same time.

The metric can be written as:

ds2 =a[η]2 ( 1+2Φ [~x]+ g [η]) dη2 − 0 0 κ + 1+2Φ [~x]+ g [η]+ r2 dr2 + r2 (1+2Φ [~x]+ g [η]) dΩ2 + ... (6.7) 0 1 6 0 1 i Any other second order corrections to the metric, which survive ensemble averaging,

will result in a universe that is not statistically invariant under rotations and trans-

lations. g0[η] can be thought of as containing a second order gauge transformation of the time parameter, while g1[η] can be thought of as the second order correction to the scale factor. Finally, κ is the spatial curvature4, with κ < 0 corresponding to

4Explicitly it is the Ricci scalar on the spatial subspace in the homogeneous and isotropic limit. Chapter 6: Perturbing an FLRW Universe 77

negative curvature, κ = 0 corresponding to a ﬂat spacetime, and κ> 0 corresponding

to positive spatial curvature. The most general form of the ﬂuid 4-velocity is then

given by:

1 3Φ [~x]2 η2(∂ Φ [~x])2 1 4 U µ = 1 + Φ [~x]+ 0 + i 0 + g [η], ∂ Φ [η, ~x] (6.8) mat a[η] 0 2 18 2 0 (0) j 0 ( ηρmat ) The stress energy tensor is then given by:

T µν =ρ U µ U ν Λgµν mat mat mat − ρ(0) 1 where ρ = mat 1+2Φ[~x] η2 2Φ [~x]+ δ(2) (6.9) mat a[η]3 − 6 ∇i 0 mat Solving the Einstein equations, we are left with:

a[η]κ a[η]3Λ 97η2(∂ Φ [~x])2 g [η]=δ(2) + 4Φ [~x]2 + i 0 0 mat − (0) (0) − 0 108 2ρmat ρmat 10η2(∂ Φ [~x])2 g [η]= i 0 + c (6.10) 1 − 27 1

Here, c1 is an arbitrary constant which is closely related to a shift in the time parameters in a similar way as the constant η0 deﬁned in the ﬁrst part of equation (6.1).

Since this constant is arbitrary, no physical observable should depend on it. This will be used as a consistency check.

6.2 Shape of the Power Spectrum

At times suﬃciently long after equal matter and radiation, there is no dynamic pressure (this applies even in the case that a cosmological constant is included nonpertur- Chapter 6: Perturbing an FLRW Universe 78

batively). Since there is no dynamic pressure, the shape of the metric power spectrum

(P [k]) becomes constant in time. In the case that there is no dark energy, the amplitude is also a constant. The metric power spectrum can be split up into two parts.

ns 1 First, the primordial power spectrum (A k − ) which arises from the physics of k0

9 the early universe, with scalar ﬂuctuation amplitude A = 2.4 10− (evaluated at × 1 k = k = 0.002Mpc− ) and n is the scalar spectral index with n 0.96 [4]. We 0 s s ≈ also introduce a transfer function (T ) which takes the primordial power spectrum from the start of radiation domination to the current observable power spectrum. In the case that baryons and dark energy are ignored, the full power spectrum can be written as:

dk Φ [η, ~x]2 = P [k] 0 k Z ns 1 k − where P [k]=A T [k]2 (6.11) k 0 The transfer function can be approximated by (See equation (29) of [69] for the

approximate functional form :):

2 keq T [k] 2 2 (6.12) ∼ keq + k

1 k < keq  (6.13) ∼ keq  k2 k > keq

During radiation domination, matter perturbations could not grow. Once matter and radiation fell out of equilibrium, superhorizon modes k < keq in the matter

power spectrum grew, whereas subhorizon modes oscillated. At times much later Chapter 6: Perturbing an FLRW Universe 79

than equal matter and radiation, one ﬁnds that the matter power grows like k4 for

k < keq and oscillates for k > keq. The metric perturbation power spectrum is related

to the matter power spectrum by a factor 1 .5 ∼ k4 Ultimately, this power spectrum will be used to calculate integrals, which appear in a number of expressions including the averaged luminosity distance redshift relation

(see section 10.1), of the form:

ns 1 n dk k − k IP [n] A T [k]2 ≡ k k H Z 0 0 ns 1 n dk k − k IP [n,x] A T [k]2 sin[kx] sin ≡ k k H Z 0 0 ns 1 n dk k − k IP [n,x] A T [k]2 cos[kx] cos ≡ k k H Z 0 0 ns 1 n dk k − k IP [n,x] A T [k]2 sinintegral[kx] (6.14) sinintegral ≡ k k H Z 0 0 Here, n is associated with the number of spatial derivatives that have been taken in

the calculation, and x will be proportional to the diﬀerence in conformal time between

the source and the observer η η . The factors of H have been included so as obs − source 0

to nondimensionalize the above expressions. For ns . 1, one can approximate:

n ns 1 A kIR kIR − n< 1 n n+ns 1 H0 k0 s  − − n ns 1 IP [n]  4A keq keq − 5 n >n> 1 n  (n+ns 1)(5 n ns) H0 k0 s s ∼  − − − − −  ns 1 n 4 A kUV − kUV kUV n> 5 ns ns+n 5 k0 H0 keq −  − If we make the approximations that k k

5Transfer functions are discussed in more detail by Eisenstein and Hu in [69, 68]. See also the work by Bardeen and collaborators in [17]. Chapter 6: Perturbing an FLRW Universe 80

n A kIR n< 0 n H0 −  ns 1  A kIR − n =0  1 ns k0  − IP [n]  n (6.15) ∼  k  A eq 4 n> 0  H0 ≥ n 4 4  A kUV − H0 n> 4  H0 keq   In this instance only terms with a positive power of kUV and a negative power of kIR

are retained. In each case that a positive power of kUV survives, that term has a UV divergence, and in the case that a negative power of kIR survives, that term has a IR divergence. The remaining terms are divergence free, and one ﬁnds that the largest n terms leading contribution is similar to A keq . We can see that the largest terms H0 in a perturbative expansion comes from the largest n. This statement is only true at

times after equal matter and radiation, and can be seen by considering the scale that

the power spectrum cuts oﬀ in comparisson to the Hubble rate at that time. In the

6 early universe, H0 > kcutoff and we ﬁnd that that IP [n] > IP [n+1]. At times close to

equal matter and radiation, H k k , and we ﬁnd that IP [n] IP [n + 1]. 0 ∼ cutoff ∼ eq ∼ In general one might conclude that long wavelength ﬂuctuations have decreasing

importance, and short wavelength ﬂuctuations have increasing importance. Moving

forward we can approximate the oscillating functions:

n ns A kIR kIR k0x n 1 n+ns H0 k0 ≤−  n+1 keq IPsin[n,x]  A H0x 3 n> 1 & keqx 1 (6.16) ∼  H0  ≥ − ≪  0 4 n> 1 & k x & 1 ≥ − eq   6  Here kcutoff does not necessarily coincide with keq until equal matter and radiation. It could depend on the scale of reheating or alternatively some scale from the very early universe. Chapter 6: Perturbing an FLRW Universe 81

In the above expression, terms with possible IR divergences are picked out in the ﬁrst

expression. In the second expression, one expands the expression to leading order

in x (sin[kx] kx when k x 1). In the ﬁnal expression, one makes use of the ≈ max ≪ Riemann Lebesgue lemma.

Later, we will see that usually x = η η . Moving this to redshift space, | | | obs − source| one has:

1 1 da − dz − δη z | |≈ dη da η=ηobs ! 2 a[η] z ≈ a′[η] η=ηobs

z δη | |≈H0 The oscillating functions are then given by:

n ns A kIR kIR k0x n 1 n+ns H0 k0 ≤−  n+1 keq H0 IPsin[n,x]  A H0x 3 n> 1 & x (6.17) ∼  H0 keq  ≥ − ≪  0 4 n> 1 & x & H0 ≥ − keq    IP [n] n 0  ≤ H IPcos[n,x]  IP [n] 4 n> 1 & x 0 (6.18) ∼  ≥ − ≪ keq  H0 0 4 n> 1 & x & k  ≥ − eq    n ns A kIR kIR k0x n 1 n+ns H0 k0 ≤−  n+1 keq H0 IPsinintegral[n,x]  A H0x 3 n> 1 & x (6.19) ∼  H0 keq  ≥ − ≪  π x IP [n] 4 n> 1 & x & H0 2 x ≥ − keq  | |   Chapter 6: Perturbing an FLRW Universe 82

H0 2 We see that the oscillating functions are only important for z < 10− . The keq ∼ 2 nearest supernova used as distance indicators are found at z 10− , which also ∼ corresponds roughly to the scale of homogeneity. In chapters 10 and 11, we will make use of the approximations (6.17,6.18,6.19) when ﬁtting our model to supernova data.

6.3 Evolution of Radiation Temperature in a Per-

turbed Universe

During late times, the radiation content of the universe is small, and does not interact with the matter field. As such, it will not significantly affect the geometry, and must obey a continuity equation.

T µν =0 (6.20) ∇µ rad

In the above expression, the stress energy tensor of the radiation ﬁeld is deﬁned as:

4 1 T µν = ρ U µ U ν + ρ gµν rad 3 rad rad rad 3 rad

As was the case when considering the matter density of the universe, one has:

ρ(0) ρ = rad 1+ δ [η, ~x]+ δ(2) [η] rad a[η]4 rad rad 1 1 3 1 U µ = 1 + Φ [~x]+ g [η]+ (Φ [~x])2 + (∂ u [η, ~x])2,∂ u [η, ~x] rad a[η] 0 2 0 2 0 2 i rad j rad Putting this all together, we ﬁnd that: Chapter 6: Perturbing an FLRW Universe 83

urad[η, ~x] =0

δrad[η, ~x] =4Φ0[~x]

20η2(∂ Φ [~x])2 δ(2) [η]=δ(2) [0] + i 0 (6.21) rad rad 27

2 Here, δrad[η] is simply an homogeneous isotropic correction to the radiation density of the universe.

If one assumes that the radiation ﬁeld is well approximated by a blackbody spectrum,

one ﬁnds that the temperature of the radiation background (CMB) as measured by

observers comoving with the matter ﬁeld is given by:

1/4 α β µν TCMB[η, ~x] =¯σ gµαgνβUmatUmatTrad (6.22)

2 (2) T0 3 2 2η 2 δrad[0] = 1 + Φ0[~x] Φ0[~x] + (∂iΦ0[~x]) + (6.23) a[η] − 2 9 4 !

In the above expression, the temperature of the CMB is related to the fourth root of

the energy density of the radiation ﬁeld by constant (¯σ) which is closely related to

the Stefan-Boltzmann constant.

6.4 Geodesics and Comoving Observers, Light Cones

and Light Cone Angles

In this section we introduce some scalar quantities.

The ﬁrst is the quantity τ which has multiple purposes in our work, namely; it can Chapter 6: Perturbing an FLRW Universe 84

be used as a clock to define spatial averaging7, it can be used to define a congruence of geodesic observers (U µ = gµν∂ τ), and finally if all observers comoving with the − ν matter field are geodesic, it also defines a congruence of comoving observers.

Next we can introduce the null analogue ω. Given a particular observer, the past light

cone of the observer is the subspace deﬁned by ω = 0, and the congruence of null

µ µν geodesics k = g ∂µω. Once the past light cone has been deﬁned, we can write two

angles which are conserved on the past light cone namely θ¯a where the index a labels the two light cone angles. We will argue that these scalar quantities coincide with the coordinates of the geodesic light cone gauge introduced in [76]. Figure 6.1 shows a spacetime diagram highlighting the importance of these diﬀerent scalar quantities.

We begin by studying the quantity τ.

∂xµ uµ ≡ ∂τ = gµν∂ τ − ν uµu = 1 µ − uµ uν =0 (6.24) ∇µ

The quantity τ is a scalar quantity and is therefore invariant. If we choose to work

in a synchronous gauge, then τ is the temporal co-ordinate.

We already deﬁned the stress energy tensor of a particular ﬂuid8 as:

7See section 8.4. 8Specifically we consider a fluid with no anisotropic stress as measured by an observer comoving with the fluid. Chapter 6: Perturbing an FLRW Universe 85

Tµν =(ρ + P )UµUν + Pgµν (6.25)

If Uµ = uµ from equations (6.24) and (6.25), then observers comoving with the ﬂuid are geodesic observers. In a matter dominated universe, such as we are considering in this model, comoving observers are geodesic observers.

The surface of constant τ picks out a space-like hypersurface of spacetime where all geodesic observers will measure the same rates on their clocks.

In a generic FLRW universe with linear perturbations such as that given in equation (6.2), we can write:

η τ[xµ]= dξa[ξ](1 Φ[ξ, ~x]) + ... − Zη0 In the above expression, τ is deﬁned everywhere in spacetime with reference only to some arbitrary initial “time” η0.

In the spacetime defined in section 6.1, one finds that the τ defined as comoving with the matter field is geodesic9, and can be written as:

η3ρ(0) 3 17η2 τ[η, ~x]= mat 1 Φ [~x]+ Φ [~x]2 (∂ Φ [~x])2 36 − 0 2 0 − 72 i 0 2 6 (0) κη2 1 η Λ ρmat + δ(2) (6.26) 80 − 2 mat − 10368    In a similar way, a null scalar ω can be used to deﬁne a congruence of null 4-velocities:

9Since matter is pressureless and noninteracting, this result is as expected. Chapter 6: Perturbing an FLRW Universe 86

∂xµ kµ ≡ ∂ω = gµν∂ ω − µ µ k kµ =0 (6.27)

We usually consider photons which have not interacted since they were emitted, and as such move along null geodesics. Here a congruence of null geodesics centered at a particular observer is considered. We can write any null vector which passes through the observer as kµ10, which obeys:

kµ kν =0 (6.28) ∇µ

The surface deﬁned by ω = 0 picks out the past light cone of a particular observer.

Unlike τ, which is deﬁned generally, the deﬁnition of ω makes reference to some particular observer, as well as the position of a source. For an observer on the apex of the past light cone, and a source somewhere else on the past light cone, in a generic

FLRW universe with linear perturbations such as that given in equation (6.2), we can write:

10Here kµ is a null 4-velocity, which should not be confused with ~k used to deﬁne the linear perturbations in Fourier space. The distinction between them should be clear from the use of a Greek index on the null 4-vector kµ. Chapter 6: Perturbing an FLRW Universe 87

We can see that when integration along the line of sight is needed, general perturbative

expressions become more complicated. In order to progress, we will also make explicit

use of the geometry deﬁned in section 6.1, which is simpliﬁed based on the fact that

the metric perturbation is time independent. The Fourier transform introduced in

equation (6.4) will also be used when convenient, and ﬁnally a shorthand will be

adopted to improve the ease of reading. We can deﬁne:

~r = ~x ~x source − obs r = ~x ~x | source − obs| k = ~k (6.29)

For an observer situated at η = η0 and ~x = ~xobs in the geometry deﬁned in section 6.1, using the Fourier transform deﬁned in equation (6.4), and the shorthand introduced in equation (6.29), the null scalar can be written as: Chapter 6: Perturbing an FLRW Universe 88

ω[xµ ,xν ]=r + η η obs source source − obs 3 √ d k i 2r i~k ~xobs i~k ~xobs + e · e · φ0[k]E[~k] (2π)3/2 ~k ~r − Z · 2 (0) (2) 7 7 δ Λ ρmat (ηobs ηsource) + mat (η η )+ − 2 obs − source 24192 κ + η3 η3 4rη2 4r2η 48 obs − source − source − source 3 2 3 d k 4k r ~ 4r ~ + 3 P [k] 1 cos[k ~r] + cos[k ~r] 1 4πk " (~k ~r)4 { − · }! (~k ~r)2 { · − }!# Z · · 3 2 3 2 d k k r 19k r 2 2 + P [k] 2r + r +3rηsource +3η 4πk3 − ~ 2 216 source Z " (k ~r) # · (6.30)

Since this geometry is statistically invariant under rotation, we can choose to simplify

the problem by rotating the co-ordinate system. This point is discussed further in

section 8.7.2. If we are performing the integral over both the angles in Fourier space

as well as the angles in real space, we can rotate our system such that θ measures the

angle between ~k and ~x so that we can write ~k ~x = ~k ~x cos(θ), where θ is simply · | |

the zeroth order coordinate deﬁned in the deﬁnition of the metric11.

The null coordinate can then be written as:

11This point is subtle, and can only be performed if we are in a statistically isotropic spacetime, and we are averaging over the spatial angles. Chapter 6: Perturbing an FLRW Universe 89

ω[xµ ,xν ]=r + η η obs source source − obs ~ d3k 2ieik ~xobs + · 1 eikr cos[θ] φ [k]E[~k] (2π)3/2 k cos[θ] − 0 Z 2 (0) (2) 7 7 δ Λ ρmat (ηobs ηsource) + mat (η η )+ − 2 obs − source 24192 κ + η3 η3 4rη2 4r2η 48 obs − source − source − source dk 4 + P [k] 1 cos[kr cos[θ]] k k2r cos[θ]4 { − } Z dk 4 + P [k] cos[kr cos[θ]] 1 k k2r cos[θ]2 { − } Z dk r 19k2r + P [k] 2r + r2 +3rη +3η2 k − cos[θ]2 216 source source Z (6.31)

In general, the above expression comes with an integration constant. This integration constant is ﬁxed by the choice that the apex of the past light cone coincide with the position of the observer. Alternatively stated, the co-ordinate system is unperturbed in the neighborhood of the observer.

We can also deﬁne “light cone angles” θ¯a (where a is an index that takes on two values) as angles that remain constant along null geodesics centered at a particular observer:

kµ θ¯a 0 (6.32) ∇µ ≡

In particular, for an observer situated at η = η0 and ~x = ~xobs in the geometry deﬁned in section 6.1, using the Fourier transform deﬁned in equation (6.4), the shorthand Chapter 6: Perturbing an FLRW Universe 90

introduced in equation (6.29), and a rotation to align the positive z axis with ~k, we

have:

3 1 d k i sec[θ]tan[θ] i~k ~x θ¯ =θ +2 φ [k]E[~k] tan[θ] e · obs (2π)3/2 0 kr − Z 3 d k i sec[θ]tan[θ] i~k ~x +ikr cos[θ] +2 φ [k]E[~k] e · obs (2π)3/2 0 kr Z dk 4sin[θ] + P [k] 4 2cos[θ]2 k2r2 cos[θ]2 + k2r2 cos[θ]4 k k2r2 cos[θ]5 − − Z dk 4sin[θ] + P [k] 2cos[θ]2 cos[kr cos[θ]] 4cos[kr cos[θ]] krx sin[kr cos[θ]] k k2r2 cos[θ]5 − − Z θ¯2 =ψ (6.33)

We notice that the functional form of θ¯1 and θ¯2 are very diﬀerent. This diﬀerence can

be accounted for by considering the rotation of the coordinate system. In general,

∂ ~x ~k = 0, and the symmetry between θ¯1 and θ¯2 is restored. However, we choose ∂ψ · 6 to rotate our coordinate system in such a way that ~x ~k = ~x ~k cos[θ], in which · | | case one has ∂ ~x ~k = 0 leading to the expressions given above. Once again this ∂ψ · rotation can only be performed if we are ultimately performing the integration over the angles in both Fourier space and real space.

As was the case for the expression for ω, there are integration constants that are chosen in such a way that the coordinate system coincides with the unperturbed coordinate system in the neighborhood of the observer.

6.4.1 Geodesic Light Cone Gauge

The geodesic light cone gauge (GLC) is deﬁned in [76] using the coordinates τ,ω, θ¯a used above. The metric is given by: Chapter 6: Perturbing an FLRW Universe 91

ds2 =Υ2dω2 2Υdωdτ + γ dθ¯a U adω dθ¯b U bdω − ab − − In particular, we can calculate the quantity:

ab µν a b γ = g ∂µθ¯ ∂µθ¯ (6.34)

This expression will be used together with the expression for the area distance found

in GLC gauge using exact Jacobi maps in [73] to ﬁnd a perturbative expression for

the area distance.

Finally, we can deﬁne the redshift with respect to a particular observer as:

λ 1+ z = obs (6.35) λsource µ k Uµ α α |x =xsource = ν (6.36) k Uν xα=xα | obs

In particular, for an observer situated at η = η0 and ~x = ~xobs in the geometry deﬁned in section 6.1, we have: Chapter 6: Perturbing an FLRW Universe 92

2 ηobs 1+ z = 2 (6.37) ηsource 2 3 ηobs d k i~k ~xsource ikηsource cos[θ] + φ [k]E[~k]e · 1+ η2 (2π)3/2 0 3 source Z 2 3 ηobs d k i~k ~x ikηobs cos[θ] φ [k]E[~k]e · obs 1+ − η2 (2π)3/2 0 3 source Z 2 2 2 2 ηobs κ(ηobs ηsource 2rηsource r ) + 2 − − − ηsource 12 dk 2η2 + P [k] obs (1 cos[kr cos[θ]]) k 3rη cos[θ]2 − Z source dk 2(3r η )η2 kr cos[θ] + P [k] − source obs sin2 k 3rη2 2 Z source dk (57r2 + 114rη + 109η2 109η2 )η2 + k2P [k] source source − obs obs k 216η2 Z source dk cos[θ]2(η η cos[kr cos[θ]])η3 + k2P [k] obs − source obs (6.38) k 9η2 Z source Chapter 6: Perturbing an FLRW Universe 93

Figure 6.1: Spacetime diagram showing surfaces of constant τ and ω as well as in-

dicating one of the light cone angles. We can see that based on this diagram, using

τ, ω and θa as a coordinate system would be convenient when studying observables on the past light cone. Chapter 7

Jacobi Maps and the Area

Distance in a Perturbed Universe

In a universe that is not perfectly FLRW, deﬁning the luminosity distance is not sim-

ple. Fortunately, there is a technique for calculating the area distance on a manifold

using Jacobi maps. Here the analysis by [73] will brieﬂy be summarized.

µ We can deﬁne two vectors sA (labeled by an uppercase latin index) which satisfy:

µ ν gµνsAsB =δAB

µ UµsA =0

µ kµsA =0

Πµkα sν =0 ν ∇α A µ µ µ µ µ k kν k Uν + U kν where Πν =δν α 2 α − (U kα) − U kα

µ µ Here, Πν projects a two-dimensional space of vectors orthogonal to U , and the space-

94 Chapter 7: Jacobi Maps and the Area Distance in a Perturbed Universe 95

like 4-vector nµ, which points along the line of sight of the null geodesic kµ.

µ µ α 1 µ n =U +(U kα)− k

µ The vectors sA are labeled with upper case Latin indices (e.g. A) where the position of the index is unimportant. Using these expressions, we can then write down the deﬁning relationships:

R R kαkγsβ sσ AB ≡ αβγσ A B µ ν A A C k ∂µk ∂νJB =RC JB

A µ µ JB (xo ,xo ) =0

µ A µ µ µ k ∂µJB (xo ,xo ) =(k uµ)oδAB (7.1)

µ µ Here, Rαβγσ is the Riemann curvature tensor, and JAB = JAB(xs ,xo ) is the Jacobi

µ µ map, which connects the observer situated at xo to the source situated at xs . The

last two relations are simply initial conditions. The Jacobi map is related to the area

distance by1:

2 A dA =det JB (7.2) Performing this calculation in the GLC gauge, with the observer at the origin (see

[73] for details), we can show that the area distance is given by:

1This expression is discussed in chapter 4 of [67]. Chapter 7: Jacobi Maps and the Area Distance in a Perturbed Universe 96

µ k U˜ √γsource d2 = µ A ν 1 k Uν sin[θ¯ ]

Here, U˜ µ is the 4-velocity comoving with the observer and U µ is a geodesic passing

through the observer. As discussed in section 6.4, 4-velocities comoving with the

µ k U˜µ matter ﬁeld are geodesic in the geometry of interest to us. In this case, ν = 1. k Uν

1 1 The factor of sin[θ¯ ]− related to ﬁxing the initial condition given by the last equality

in equation (7.1), and γsource is a shorthand for:

µν ¯a ¯b γsource = det g ∂µθ ∂νθ µ µ x =xsource The area distance is then given by:

γ 1/4 d = s (7.3) A sin[θ¯1]2

In particular, for an observer situated at η = η0 and ~x = ~xobs in the geometry deﬁned in section 6.1, we have:

2 3 rη Ω d k 1 2i i~k ~x d = (1 + φ [k]E[~k] 1+ + e · obs A 12 (2π)3/2 0 cos[θ]2 kr cos[θ]3 Z 2 3 rη Ω d k 1 2i i~k ~x +ikr cos[θ] + φ [k]E[~k] e · obs 12 (2π)3/2 0 cos[θ]2 − kr cos[θ]3 Z rη2Ω dk 28 cos[kr cos[θ]] 28 4sin[kr cos[θ]] + P [k] − + 12 k k2r2 cos[θ]6 kr cos[θ]5 Z rη2Ω dk 28 28 cos[kr cos[θ]] 9 + cos[kr cos[θ]] 10 sin[kr cos[θ]] + P [k] − + 12 k k2r2 cos[θ]4 cos[θ]4 − kr cos[θ]3 Z rη2Ω dk cos[kr cos[θ]] 5 1 5k2η2 + P [k] − (7.4) 12 k cos[θ]2 − 2 − 27 Z Chapter 7: Jacobi Maps and the Area Distance in a Perturbed Universe 97

This expression agrees with the expressions given in various papers by Gasperini et

al [73, 58, 21, 20]2.

We see that long wavelength (k 0) ﬂuctuations are not sensitive to the orientation of →

the structure, whereas the short wavelength ﬂuctuations (k & H0) are. In particular,

we ﬁnd that for short wavelength ﬂuctuations the correction to the luminosity distance

due to structure forming along the line of sight of the observer is relatively small,

whereas the corrections to structure forming transverse to the line of sight are larger.

This seems to suggest that it is anisotropies and not inhomogeneities that cause the

largest corrections to the luminosity distance redshift relation.

The second order corrections are of the form:

(2) IR ∆ dA 1 FLRW − IP [0] dA ≈ 2

5 2 (2) UV − (H η ) IP [2] θ 0 ∆ dA 27 0 source ∼ dFLRW ≈  A  13 (H η )4IP [4] θ π  360 0 source ∼ 2 It is premature to draw these conclusions since η is not observable and the angle θ is not conserved on the past light cone beyond zeroth order. One might also argue that the energy density at the observer is perturbed, and therefore might correspond to an advanced or a delayed expression for the area distance.

To conclude we must realize that since the luminosity distance is not written in terms of physically meaningful variables, the above statements are no more than an indication that IR corrections are not sensitive to the orientation of structure and that

2While they do not give explicit expressions for the same background geometry that we have used here, the formulae used for the luminosity distance is in agreement. The expression for the luminosity distance up to ﬁrst order is in explicit agreement with [73, 58, 21, 20]. Chapter 7: Jacobi Maps and the Area Distance in a Perturbed Universe 98

UV corrections are more sensitive to anisotropies than they are to inhomogeneities. Chapter 8

Averaging Techniques and

Consequences at Late Times

In this chapter we will discuss a formalism for averaging scalar observables. We begin with a discussion on the distinction between geometric and ensemble averaging.

We then define geometric averaging of scalar quantities on a manifold, and, more interestingly, we can define restricted scalar averaging on a submanifold defined by scalar constraints. Three types of geometric averages are then described, namely spatial averaging, light cone averaging, and combined spatial and light cone averaging.

Spatial averaging can be defined for any observable that is defined locally on the spacetime manifold. The average is then taken over all similar observers, where similar observers are defined by those observers who measure the same value for a

“clock” field. This clock field is an observable that is defined locally for any observer, and is not unique. In this work three different clock fields are considered: the temporal

99 Chapter 8: Averaging Techniques and Consequences at Late Times 100

co-ordinate as defined by choosing synchronous gauge 1, a subdominant entropy field2, and the energy density of the dominant field (matter). Although there are many other valid alternative clock fields3, we will not consider them here. Light cone averaging can be defined for any observables that are dependent on both a source and an observer. The average is then taken over each of the possible sources associated with a particular observer[76]. Finally, combined spatial and light cone averaging can be defined for any observable that is dependent on both a source and an observer, in which case the average is defined over each source and observer pair.

8.1 Ensemble and Geometric Averaging

There are two diﬀerent types of average that need to be accounted for.

If we use an analogy from statistical mechanics, we can define a number of large thermal systems which have different microscopic realizations, but which are indis- tinguishable on large scale. Similarly, we could imagine a number of universes which, while microscopically different, give rise to the same large scale observables. Just as we can argue for taking an ensemble average over all microstates in statistical mechanics, we can argue for taking an ensemble average over all realizations of the universe. Since the exact geometry of the universe is unknown, ensemble averaging is used to average over the ignorance of small scale structure in our universe. This so called ensemble averaging was discussed in section 6.1.2. It is worth noting that

1We also show that this choice is equivalent to Buchert averaging[39] 2speciﬁcally the temperature of the CMB 3See for example the constant mean curvature foliation as discussed in [29], which is closely related to the uniform Hubble ﬂow gauge of [16]. Chapter 8: Averaging Techniques and Consequences at Late Times 101

while ensemble averaging is used, ergodicity[10] is not assumed4.

We also consider an average that is defined on a particular geometry. Since many observables are dependent on the geometry near to an observer, we find that, in general, different observers may see different values for the same observable in the same universe. Geometric averaging will be used to average over the same observable in the same universe for different observers and, in some cases, differing sources.

Ensemble averaging can be thought of as an average over the geometry of the universe, whereas geometric averaging can be thought of as an average on a particular geometry.

2 In this thesis, ensemble averaging is denoted by an overline (e.g. Φ0[~x] ) whereas geometric averaging is represented by angular brackets with the subscript denoting the constraints used to deﬁne the embedded subspace (e.g. ρmat ) h iχ=χ0

8.2 Embedded Subspaces

In a perfect FLRW universe, we can easily separate spacetime into space and time since space is a maximally symmetric subspace of spacetime. In a perturbed universe, however, such a separation is not as simple. In order to proceed, we must ﬁnd a way of picking out a subspace of spacetime. What follows is motivated by the lecture notes

4Ergodicity is a difficult constraint to impose on a theory beyond first order in perturbation theory. In particular, as we shall show in the remainder of this chapter, spatial averaging is sensitive to the clock used to define constant time hypersurfaces. Different clocks therefore lead to different spatial averages. In order to define a clock for which the spatial average of the energy density of the matter field corresponds to the ensemble average of the energy density of the matter field, one must enforce particular constraints on the clock field. In particular, one finds that the first order correction to the clock field must vanish in any gauge in any geometry. If such a scalar exists, it is difficult to interpret physically. This point will be discussed further in chapter 9 Chapter 8: Averaging Techniques and Consequences at Late Times 102

written by Matthias Blau[31]5, in which he discusses a d 1 dimensional subspace of − a d dimensional Riemannian space.

Consider a d dimensional Riemannian6 manifold with metric g. We can deﬁne M a d n dimensional submanifold ∂ of using n linearly independent injective − M M scalar constraints Qi =07.

We can deﬁne an arbitrary vector normal to ∂ as N µ, and an arbitrary vector M transverse to the subspace as T µ. A tensor that projects vectors onto the subspace

can be written as γαβ and must obey:

α γµαγν =γµν

α β γβ N =0

α β α γβ T =T

and γα =d n (8.1) α −

iµ µν i We can deﬁne a basis of vectors normal to the subspace by N = g ∂νQ , where

lower case Latin indices beginning at i (e.g. i,j,k,... ) label the n vectors in the basis.

aµ µν a Similarly a basis of vectors tangent to the subspace can be deﬁned by T = g ∂νy

where lower case Latin indices starting at a (e.g. a,b,c,... ) label the d n vectors − in the basis. ya can be used as co-ordinates on the subspace.

5See also numerous works on the ADM formalism of general relativity [13](including citations therein), and more recent reviews [83]. 6This analysis applies to pseudo Riemannian manifolds as well. 7Linear independence of the constraints ensures that the scalar quantities are not redundant (z = 0 and z2 = 0 are the same constraint). Multivalued constraints may leave ambiguities about which subspace must be used. For example, z2 1 = 0 has multiple solutions so, unless z > 0 is implied, it is not well deﬁned. − Chapter 8: Averaging Techniques and Consequences at Late Times 103

We can then make the ansatz that the projection tensor can be written as8:

i1 µ1 j1 in−1 µn−1 jn−1 in jn j1 nǫi1,...,in ǫi1,...,in ∂µi Q ∂ Q ...∂µn−1 Q ∂ Q ∂αQ ∂βQ Q γαβ =gαβ k1 ν1 l1 kn νn ln − ǫk1,...,kn ǫl1,...,ln ∂ν1 Q ∂ Q ...∂νn Q ∂ Q (8.2)

We can choose to work in a gauge with coordinates:

Xµ = Qi,ya (8.3) In the above gauge choice, Qi are the constraints, and ya are the coordinates on the

µi µν i subspace. In this basis, the vectors g = g ∂νQ form a basis of vectors normal to

µa µν a the subspace, and the vectors g = g ∂νy form a basis of vectors tangent to the

subspace. The projection operator can then be written as:

i1 µ1 j1 in−1 µn−1 jn−1 in jn nǫi1,...,in ǫi1,...,in ∂µ1 Q ∂ Q ...∂µn−1 Q ∂ Q ∂αQ ∂βQ γαβ =gαβ k1 ν1 l1 kn νn ln − ǫk1,...,kn ǫl1,...,ln ∂ν1 Q ∂ Q ...∂νn Q ∂ Q

i1 µ1j1 in−1 µn−1jn−1 in jn nǫi1,...,in ǫi1,...,in δµ1 g ...δµn−1 g δα δβ =gαβ k − 1 ν1l1 kn νnln ǫk1,...,kn ǫl1,...,ln δν1 g ...δνn g i1j1 in−1jn−1 in jn nǫi1,...,in ǫi1,...,in g ...g δα δβ =gαβ k1l1 knln − ǫk1,...,kn ǫl1,...,ln g ...g

ij 1 in jn =g (g )− δ δ αβ − injn α β =g (g g habg )δi δj αβ − ij − ia jb α β ab =gαah gβb (8.4)

8This deﬁnition works provided that if, a single constraint is used, it is not a null constraint. In the case that a single null constraint is considered, we would need to introduce an additional vector α α Nˆ such that Nˆ ∂αQ = 0. 6 Chapter 8: Averaging Techniques and Consequences at Late Times 104

ab 1 Here, h is the inverse of the metric on the subspace ((gab)− ). It is worth noticing that:

hab =gab

ab h =Inverse[hab]

=gab = gaαg gβb 6 αβ

We can show that in this gauge γαβ obeys all the relevant properties of a projection

operator as given in equation (8.1). Since this object is a tensor, it is equally well

deﬁned in any other basis. The interval on the subspace can be written as:

2 a b ds ∂ habdy dy | M ≡ µ ν =gµνdx dx ∂ | M α β µ ν λ σ =γµ γν γσ γλgαβdx dx

α β =γαβdx dx

In general, one can deﬁne the pullback operator by:

∂xµ (8.5) ∂ya

In this case we can write the metric on the subspace as:

∂xµ ∂xν h = g ab µν ∂ya ∂yb Chapter 8: Averaging Techniques and Consequences at Late Times 105

A block matrix A must obey the following properties.

A11 A12 A =   A21 A22     1 1 1 1 1 (A11 A12A− A21)− A− A12(A22 A21A− A12)− 1 − 22 − 11 − 11 A− =   ⇒ 1 1 1 1 1 A22− A21(A11 A12A22− A21)− (A22 A21A11− A12)−  − −    (8.6)

1 and det[A] =det[A ]det[A A A− A ] (8.7) 11 22 − 21 11 12

Using the above properties, we can ﬁnd the determinant of the full metric in terms

of the determinant of the metric on the subspace. In this case we have:

det[gµν] =det[gij]det[gab gaig gbj] − ij ab ai bj 1 also (g g g g )− =g = h − ij ab ab det[gij] det[gµν]= det[hab]

det[h ] =det[gij]det[g ] ⇒ ab µν

In the ﬁrst line we used the deﬁnition of the determinant from equation (8.7), in the second line we used the expression for the inverse of a block matrix from equation

1 1 (8.6). In the last line we have made use the property det[A− ] = det[A]− .

More generally, we can write the metric on the subspace as:

i α j det[hab] = det[∂αQ ∂ Q ]det[gµν] (8.8) Chapter 8: Averaging Techniques and Consequences at Late Times 106

8.3 Restricted Geometric Averaging

Different observers will, in general, see different temperatures for the CMB. As a toy model, we consider three different observers, one on the earth today, one near the center of the galaxy, and one on the earth at the time the earth was formed (about four billion years ago). Since photons will be blueshifted as they fall toward the center of the galaxy, an observer near the center of the galaxy will see a warmer CMB than an earthbound observer. Similarly, an earthbound observer four billion years ago will have seen a warmer CMB than an observer today since the universe has cooled since then.

In order to get a meaningful average, we need to deﬁne some property of the observer.

In the case of an FLRW universe, specifying the time of the measurement is sufficient because observables do not depend on spatial position. Since we want to consider a perturbed universe, things are not as clear. Take, for example, the matter density and CMB temperature of the universe. Since matter and radiation fell out of thermal equilibrium, matter has a greater tendency to clump than radiation. Since matter density and the radiation temperature are independent, one could find two observers who see the same temperature of the CMB, but different matter density.

In general, we can average scalar observables while holding other scalar observables constant. Using the example above, we might want to study the average temperature of the CMB for all observers with the same matter density, or, conversely, the average matter density of all observers who see the same CMB temperature.

In this section, a general approach to averaging over ﬁxed scalar quantities is discussed. Consider a scalar observable S deﬁned on a d-dimensional Riemannian man- Chapter 8: Averaging Techniques and Consequences at Late Times 107

ifold with metric g as well as n linearly independent constraints Qi, we can deﬁne M a d n dimensional subspace ∂ with induced metric h. − M⊂M We can deﬁne both the geometric average of an observable (S) over the entire manifold, as well as the geometric average of an observable over the submanifold.

ddx det[g] S S | | (8.9) h iM ≡ ddx det[g] R p | | d n Rd − px det[h] S S ∂ | | h i M ≡ dd nx det[h] R − p | | d n j µ k d − x det[g] det[∂ Q ∂ Q ] S = R p| | | µ | dd nx det[g] det[∂ Qj∂µQk] R − p | p| | µ | ddx det[g] det gµν∂ Qj∂ Qk δ[Qi]S = R |p | |p µ ν | i (8.10) ddx det[g] det gµν∂ Qj∂ Qk δ[Qi] R p | p| | µ ν |Q i Since the constraints Qi areR scalars,p it isp worth noting that theQ last line in the above

expression is fully covariant. It does not rely on the choice of coordinates.

In general, this expression will be complicated, and requires some sort of approxima-

tion technique. In this study, perturbation theory will be used.

8.4 Spatial Averaging and Clock Fields

In an ideal FLRW universe, one can easily separate spacetime into constant time

hypersurfaces. Once the geometry has been perturbed, diﬀerent observables will not

generally follow the same evolution. As such, using diﬀerent observables as clocks

will lead to diﬀerent embedded subspaces. We will refer to the observable used to

deﬁne the subspace as a clock. Chapter 8: Averaging Techniques and Consequences at Late Times 108

In this section, we introduce the idea of spatial averaging, and discusses its connection

to spatial averaging deﬁned by Buchert [39].

Before proceeding, the spatial average of an observable with respect to the clock O ﬁeld χ is deﬁned by:

4 µ d x√ g ∂µχ∂ χδ[χ χ0] [χ0] − − − O (8.11) hOiχ ≡ d4x√ g ∂ χ∂µχδ[χ χ ] R −p − µ − 0 R p 8.4.1 Buchert Averaging

Buchert’s averaging scheme is found by considering spacetime in a synchronous gauge:

ds2 = dτ 2 γ dxidxj − − ij

The Buchert average is then deﬁned in this particular gauge as:

d3x det[γ]S S = Buchert 3 h i R d xp det[γ] This calculation is done in a speciﬁc gaugeR withp coordinates Q = τ τ ,xi where { − 0 } Q is a constraint, and xi are the coordinates on the subspace. We can write this expression more generally as:

3 µν d x det[g] g ∂µτ∂ντS S = − − h iBuchert d3x det[g] gµν∂ τ∂ τ R p − p − µ ν We can then write: R p p Chapter 8: Averaging Techniques and Consequences at Late Times 109

dτδ[τ τ ]f[τ] f[τ ]= − 0 0 dτδ[τ τ ] R − 0 Using this expression, we can connect theR averaging formalism deﬁned by Buchert to

our averaging formalism:

4 µν d x det[g]δ[τ τ0] g ∂µτ∂ντS S = − − − h iBuchert d4x det[g]δ[τ τ ] gµν∂ τ∂ τ R p − − 0p − µ ν = RS p p h iτ=τ0

By showing that Buchert’s original formalism, which was built using a particular choice of gauge, is equivalent to a gauge invariant averaging over an embedded subspace, we have shown that Buchert averaging is gauge invariant.

8.5 Light Cone Averaging

When considering observables that depend on both the source and the observer (hereafter referred to as nonlocal observables), spatial averaging as discussed in section 8.4 is no longer applicable. Light cone averaging was described in [76]. In their approach, the authors picked a particular observer, then defined the average over the past light cone at a particular Buchert time (see figure 8.1). The average was defined as9:

d4x√ g ∂ ω∂µτ δ[ω]δ[τ τ ]S S [τ ,xµ ] − | µ | − s (8.12) h iLC s obs ≡ d4x√ g ∂ ω∂µτ δ[ω]δ[τ τ ] R − | µ | − s R 9This notation is slightly diﬀerent to that used in [76]. Chapter 8: Averaging Techniques and Consequences at Late Times 110

Figure 8.1: Light cone averaging as described in [76]. In the frame on the left the average is taken over all sources at a particular value of the Buchert time (τ) on the past light cone of a particular observer. In the frame on the right the average is taken over all sources at a particular redshift. The two diagrams show that these averaging surfaces are, in general, diﬀerent. The solid curves show an unperturbed universe and the dashed curves show a perturbed universe.

Physical observables are usually measured as a function of redshift rather than as a function of Buchert time. This point is discussed at the end of chapter 5.1 of [76], as well as in [24]. It makes more sense to define the averaging at fixed redshift rather than at fixed geodesic time. In this case the light cone average would be written as:

d4x√ g ∂ ω∂µz δ[ω]δ[z z ]S S [z ,xµ ]= − | µ | − s (8.13) h iLCz s obs d4x√ g ∂ ω∂µz δ[ω]δ[z z ] R − | µ | − s R Here we have deﬁned the redshift of a photon emitted at the source and observed at

10 the observer as zs. Unless the relative entropy between τs and zs is zero (for ω = 0),

10The idea of relative entropy will be discussed in more detail in section 8.7.4) Chapter 8: Averaging Techniques and Consequences at Late Times 111

we cannot invert τs as a function of zs after averaging, as this would reintroduce some angular dependence in a nontrivial way. We could, however, invert the expression for the average redshift:

z [τ ,xµ ] τ [ z ,xµ ] h iLC s obs → s h i obs

In this case, we would write:

S [ z ,xµ ] h iLC h iLC obs

It is worth explicitly noting that, in general, the redshift (zs) will have a diﬀerent angular dependence than τs, and as such the backreaction associated with redshift will not be the same as the backreaction associated with Buchert time. In general we will ﬁnd that:

S [ z ,xµ ] = S [z ,xµ ] h iLC h iLC obs 6 h iLCz 0 obs

In each of the above expressions, we make no mention of ensemble averages or the details of the geometry, and the statements are completely general. It is also worth noting that there is a residual observer dependence in each of the above expressions.

The diﬀerence between redshift and τ was explicitly considered in [24]. Below I will outline the details of their calculation. The calculation is performed using GLC gauge.

In this case one has: Chapter 8: Averaging Techniques and Consequences at Late Times 112

d4x√ gδ[τ τ ]δ[ω]gτωS S = − − s h iLC d4x√ gδ[τ τ ]δ[ω]gτω R − − s 2 ¯ ¯a ¯a d θ√γ[τs, ω =0, θ ]S[τs, ω =0, θ ] = R 2 ¯ ¯a R d θ√γ[τs, ω =0, θ ] This expression is then inverted to redshiftR space:

2 ¯ ¯a ¯a ¯a ¯a d θ√γ[τs[zs, ω =0, θ ], ω =0, θ ]S[τs[zs, ω =0, θ ], ω =0, θ ] S = LC 2 ¯ ¯a ¯a h i R d θ√γ[τs[zs, ω =0, θ ], ω =0, θ ] Here the usual deﬁnition of redshiftR has been used:

µ k Uµ xα=xα | obs 1+ z = ν k Uν α α |x =xsource gτω = |τ=0,ω=0 gτω |τ=τs,ω=0 1 ≡Υ

Here we have deﬁned Υ for convenience of notation. One can invert expressions by

making use of Dirac delta functions:

f[y]= δ[g[x] y]f[g[x]]g′[x]dx − Z

2 ¯ ¯a 1 ∂ 1 ¯a d θdτs√γ[τs, ω =0, θ ]δ [1 + z Υ− ] (Υ− )S[τs, ω =0, θ ] S = − ∂τs LC 2 ¯ ¯a 1 ∂ 1 h i d θdτs√γ[τs, ω =0, θ ]δ [1 + z Υ− ] (Υ− ) R − ∂τs R µ In the above expression some factors of k Uµ xα=xα that appear in both the numera- | obs tor and the denominator and commute with the integration, have not been explicitly written. Chapter 8: Averaging Techniques and Consequences at Late Times 113

In GLC coordinates, the expression for light cone averaging given in equation (8.13)

can be written as:

4 1 τω 1 ¯a d xΥ√γδ [1 + z Υ− ] δ [ω] g ∂τ (Υ− )∂ωωS[τs,ω, θ ] − S LCz = 4 1 τω 1 h i d xΥ√γδ [1 + z Υ ] δ [ω] g ∂τ (Υ )∂ωω R − − − 4 1 1 ¯a d x√γδ [1 + z Υ− ] δ [ω] ∂τ (Υ− )∂ωωS[τs,ω, θ ] R − = 4 1 1 d x√γδ [1 + z Υ ] δ [ω] ∂τ (Υ )∂ωω R − − − R

We can see these two expressions are identical, provided that τs is inverted to τs[zs, ω =

0, θ¯a] before the angular integrals are performed. Although the results are the same, it is more convenient to perform the calculation as an average over all sources at constant redshift than performing the calculation as an average over all sources at constant τs.

8.6 Combined Spatial and Light Cone Averaging

In section 8.5, light cone averaging was discussed. There are two important features that have not, as yet, been properly examined in previous work:

The redshift dependence of nonlocal observables has not been taken into account • before [24]. Although the redshift dependence has been taken into account, it

has been done so in a complicated way.

The observer dependence of nonlocal observables has not been taken into ac- • count. Chapter 8: Averaging Techniques and Consequences at Late Times 114

In this section, we demonstrate a method that combines spatial and light cone averaging in terms of physically relevant parameters.

Figure 8.2: This figure illustrates combined spatial and light cone averaging. The dotted line shows all observers who measure a particular value for the clock field, the solid lines shows that all sources related to a particular observer are on that observer’s past light cone, and finally the dashed line picks all sources at a particular redshift for each observer.

There are at least three diﬀerent approaches to this. The ﬁrst is to take the average over all sources for a particular observer, and then take the average over all observers.

[xµ ,xµ ] [z,xµ ] [z,χ ] hO obs source iLCz obs χ 0

One could also argue that the averages could be taken the other way around. Chapter 8: Averaging Techniques and Consequences at Late Times 115

[xµ ,xµ ] [χ ,xµ ] [z,χ] hO obs source ichi 0 source LCz

Since we cannot define the light cone without making reference to the observer on which it is centered, in this approach new observer dependence will be introduced after the spatial average has been performed. As such, this order is not as well defined as performing the light cone average first. In spite of the technical difficulty, these two approaches are equally well motivated, but give rise to different solutions.

Instead, we choose to perform this average in a third way, which puts the light cone and spatial average on an equal footing.

In order to do this, we construct a (3+1) (3+1) dimensional manifold with metric: ⊕

full gµν = gµν xµ=xµ gµν xµ=xµ | obs ⊕ | source

This metric will be block diagonal in a particular gauge by construction and, as such, the metric associated with sources and the metric associated with observers will be independent. The coordinates then correspond to the position of the observer and the position of the source respectively. This metric comes with a measure:

8 full 4 4 √ µ √ µ d x g = d xobsd xsource g xµ=x g xµ=x − obs − source Z Z p We can then deﬁne the combined spatial and light cone average of an observer as:

8 full µ 1/2 d x g ( det[∂µQi∂ Qj] ) i δ[Qi] χ,LC [χ0,z0] | | O (8.14) hOi ≡ d8x gfull ( det[∂ Q ∂µQ ] )1/2 δ[Q ] R p | µ i j | Q i i R p Q Chapter 8: Averaging Techniques and Consequences at Late Times 116

Here we deﬁne the constraints:

Q =χ χ 1 − 0

Q2 =ω

Q =z z 3 − 0

There are some important observations which follow from this expression:

1. This approach has been deﬁned nonperturbatively. The average is taken over

both sources and observers11.

2. There is no ambiguity in the order of averaging.

3. Averaged quantities are deﬁned in terms of redshift. Since the averages are

deﬁned nonperturbatively, redshift space distortions and other redshift eﬀects

have already been taken into account.

4. Averaged quantities are deﬁned on the past light cone. Again, this deﬁnition is

nonperturbative.

5. Averaged quantities are deﬁned in terms of a physical clock. Unless there is

zero entropy between diﬀerent choices of the clock, diﬀerent choices of the clock

will lead to diﬀerent expressions.

11Although perturbation theory will be used in order to ﬁnd expressions, the deﬁnition is nonperturbative. Chapter 8: Averaging Techniques and Consequences at Late Times 117

8.7 Performing the Calculation

At this point, we have described explicit formalisms for calculating restricted geometrical averages of both local and non-local observables. Unfortunately, performing these calculations is diﬃcult. In this section, we describe an explicit step by step method for calculating these quantities using perturbation theory. The process is similar for both local and nonlocal observables. It should be pointed out that although the formalism developed in this chapter is gauge invariant, one must pick a gauge in order to perform a calculation. Since we have already solved for the geometry of the universe in a longitudinal gauge in section 6.1, we will make use of the coordinate system when writing down explicit expressions.

8.7.1 Local Observables

In studying local observables, we adopt the following procedure:

1. Perform a gauge transformation on the background, so that χ(0) is one of the

coordinates.

2. Identify the series expansion for the observable up to second order (S = σ0 +

σ1 + σ2).

3. Identify the series expansion for the constraint (clock ﬁeld) up to second order

(χ = χ(0) + χ(1) + χ(2)).

4. Identify the series expansion for the measure on the subspace up to ﬁrst order

( det[h]= det[g] det[∂ χ∂µχ] µ (1 + δµ)). − | µ |≈ 0:0 p p p Chapter 8: Averaging Techniques and Consequences at Late Times 118

5. Perform a series expansion for the average up to second order12.

6. Perform an ensemble average on the resulting expressions. 13.

7. Set χ(0) = 0.14

8.7.2 Nonlocal Observables

In order to study non-local observables, we follow the procedure set out below:

1. Perform a gauge transformation on the background so that the background

(0) constraints Qi are some of the coordinates. This is similar to ﬁxing a geodesic

light cone gauge onto each possible observer.

2. Rotate the co-ordinate system so as to remove any φ dependence15.

3. Identify the series expansion for the observable up to second order (S = σ0 +

σ1 + σ2).

(0) 4. Identify the series expansion for the constraints up to second order (Qi = Qi +

(1) (2) Qi + Qi ).

5. Identify the series expansion for the measure on the subspace up to ﬁrst order

( det[h]= det[g] det[∂ Q ∂µQ ] µ (1 + δµ)). − | µ j k |≈ 0:0 p p p 12See equations (8.25) and (8.26) in section 8.7.3, where the second order average is written as a zeroth order average of second order quantities. 13Performing the ensemble average will remove any dependence on the position of the observer.

14 12 This amounts to performing the average over the vacuum manifold and setting η0 = (0) . ρmat q 15See section 8.7.2 for more details. Chapter 8: Averaging Techniques and Consequences at Late Times 119

6. Perform a series expansion for the average up to second order16.

7. Perform ensemble average on the resulting expressions17.

8. Average the resulting expressions over the zeroth order submanifold18.

Rotating the Coordinate System

Since our spacetime is statistically isotropic, and we are performing an average over both real space angles and Fourier space angles, we have a large amount of freedom in choosing how angles are labeled.

Consider the following integrals:

i~k ~x dθdφdθd¯ φ¯ sin[θ]sin[θ¯]e · Z In the above integral, we can choose to align the direction θ = 0 along ~k. In this case we have:

i~k ~x dθdφdθd¯ φ¯ sin[θ]sin[θ¯]e · Z = dθdφdθd¯ φ¯ sin[θ]sin[θ¯]eikx cos[θ] Z 16π2 sin[kx] = kx

16See section 8.7.3. The same expression applies to the combined light cone spatial averaging as to spatial averaging alone. 17Performing the ensemble average will remove any dependence on the position of the observer, leaving only the directional dependence of the source from the observer.

18 sin[θ] This amounts to multiplying by 2 , and integrating θ from 0 to π, and then using the zeroth order expressions for the constraints. In the case of the manifold deﬁned in section 6.1.1, this 12 12 amounts to setting r = η0 η, η0 = (0) , and η = (0) . − ρmat ρmat(1+z) q q Chapter 8: Averaging Techniques and Consequences at Late Times 120

The ﬁrst and last line can be checked against each other numerically, and ﬁnd good agreement. This goal of this section is to perform a similar rotation in the coordinate system.

Since zeroth order observables cannot have any angular dependence, averages over the 2-sphere are trivial, and derivatives with respect to angles trivially vanish.

We will choose to compare two deﬁnitions for ﬁrst order quantities.

3 d k i~k ~x Φ[η, ~x]= e · φ[η, k]E[~k] (8.15) (2π)3/2 Z 3 d k i~k ~x +ik ~x ~x cos[θ] Φ[η, ~x]= e · obs | − obs| φ[η, k]E[~k] (8.16) (2π)3/2 Z The first definition is the correct definition for first order quantities. The second definition is not in general correct, but is much simpler to work with. Provided we consider the angular average of quantities which are observed along the same line of sight up to second order (and possibly beyond), and observables do not prefer one transverse direction over the other, both definitions yield the same results.

At first using definition (8.15), we find that:

d2ΩΦ[η, ~x]

Z 3 2 d k i~k ~x = d Ω e · φ[η, k]E[~k] (2π)3/2 Z In this expression, we can choose to rotate our coordinate system in such a way that the zenith of our co-ordinate system is aligned with ~k, in which case we have: Chapter 8: Averaging Techniques and Consequences at Late Times 121

d2ΩΦ[η, ~x]

Z 3 2 d k i~k ~x +ik ~x ~x cos[θ] = d Ω e · obs | − obs| φ[η, k]E[~k] (2π)3/2 Z The last line is exactly the definition (8.16), and clearly at first order the two defini-

tions are equivalent.

It is not obvious that we can make this approximation beyond ﬁrst order. We consider

the product of two ﬁrst order quantities Φ[η, ~x] and Υ[η′, x~′] deﬁned by the regular

deﬁnition of ﬁrst order perturbations (equation (8.15)), where the events η, ~x and { }

η′, x~′ both lie on the null geodesic connecting a particular source and a particular n o observer, we can write:

2 d ΩΦ[η, ~x]Υ[η′, x~′]

Z 3 2 d k i~k ~x x~′ = d Ω e ·( − ) φ[η, k]υ[η, k] (2π)3 | − | Z Z 3 2 d k ik ~x x~′ cos[θ] = d Ω e − φ[η, k]υ[η, k] (2π)3 | | | − | Z Z If we had started with the ﬁrst order perturbations deﬁned in equation (8.16), we would have:

2 d ΩΦ[η, ~x]Υ[η′, x~′]

Z 3 2 d k ik ~x ~x x~′ ~x cos[θ] = d Ω e (| − obs|− − obs ) φ[η, k]υ[η, k] (2π)3 | | | − | Z Z 3 2 d k ik ~x x~′ cos[θ] = d Ω e − φ[η, k]υ[η, k] (2π)3 | | | − | Z Z Chapter 8: Averaging Techniques and Consequences at Late Times 122

In going from the second to the third line we made use of the fact that both x~′ and ~x lie along the line of sight between ~x and ~x , implying ~x ~x x~ ~x = obs source | − obs|− ′ − obs ~x x~ . We can see that both equation (8.15) and equation (8.16) lead to the same − ′

result.

Some observables such as the Jacobi map (and therefore the area distance) require derivatives on the two sphere. Unless one of the directions transverse to the line of sight is treated diﬀerently to the other, derivatives on the 2-sphere should be the same using either (8.15) or (8.16), after averaging.

We deﬁne the zeroth order Laplacian operator on the 2-sphere as:

1 ∂ ∂ 1 ∂2 2 = sin[θ] + △ sin[θ] ∂θ ∂θ sin[θ] ∂ψ2 At zeroth order there is no angular dependence, and we must ﬁnd that the Laplacian

on the 2-sphere must vanish. At ﬁrst order the Laplacian of a ﬁrst order quantity

looks like a boundary term, and since a 2-sphere has no boundary it must vanish. At

second order we can consider objects of the form19 :

d2ΩΦ[η, ~x] 2 Υ[η , x~ ] △ ′ ′ Z We can compare this expression using both deﬁnitions (8.15) and (8.16). Unfortu- nately calculating the above expression using equation (8.15) cannot be done easily for ~x = x~ , although a numeric calculation conﬁrms that using either expression gives 6 ′ the same result.

19 −2 Terms of the form ∂θΦ∂θΥ + sin[θ] ∂ψΦ∂ψΥ can be found by integrating this expression by parts. Chapter 8: Averaging Techniques and Consequences at Late Times 123

This result shows that provided η, ~x and η′.x~ both lie on a null geodesic con- { } ′ µ µ n o necting xsource and xobs and provided that no observables prefer one of the directions transverse to the line of sight over the other, we can define first order fluctuations

either by (8.15) or (8.16). This simpliﬁes our calculations. This simpliﬁcation has

been used in the deﬁnitions for the scalars used to deﬁne the co-ordinates of the past

light cone as well as the area distance in section 6.4 as well as chapter 7.

All results quoted in this thesis are quoted using this simpliﬁcation, but the expres-

sions were all explicitly tested numerically using both deﬁnitions (8.15) and (8.16).

When performing the calculation using equation (8.16) the angle θ has an alternative

interpretation. If structure is forming parallel to the line of sight, we ﬁnd that ~k and

~x ~x are aligned, and θ 0 or θ π. When structure is forming transverse to the − obs ∼ ∼ line of sight, ~k is perpendicular to ~x ~x , and θ π 20. − obs ∼ 2

8.7.3 Perturbation Theory

Ultimately, we would like to be able to write the expression for the full averaged quantity up to second order. Since the background manifold and submanifold are reasonably well understood, it would be preferable to be able to write a fully averaged quantity up to second order in terms of an average of some second order quantity on the background submanifold. To begin, we study a perturbative expansion on

Dirac delta functions. Once we understand this, we can examine the full averaged expression.

20Since θ is only a light cone angle at zeroth order, there will be perturbative corrections to this expression, and the interpretation cannot be taken too far. Chapter 8: Averaging Techniques and Consequences at Late Times 124

Perturbative Delta Functions

Since we will use perturbation theory, and since there will be perturbative corrections

to the constraints that deﬁne the subspace of the manifold, we will need a general

approach to dealing with delta functions of perturbative expressions. As a toy model

we consider:

dxf[x]h[x]δ[x + ǫg[x]] dxf[x]δ[x + ǫg[x]] R Rdxf[x]h[x](δ[x]+ ǫg[x]δ′[x]) dxf[x](δ[x]+ ǫg[x]δ [x]) ≈R ′ dxf[x]h[x]δ[x] dxf[x]h[x]g[x]δ′[x] dxf[x]h[x]δ[x] dxf[x]g[x]δ′[x] R + ǫ ǫ dxf[x]δ[x] dxf[x]δ[x] 2 ≈R R − R dxf[xR]δ[x] Rdxf[x]h[x]δ[x] d(fR[x]h[x]g[x]δ[x]) dxδ[x]Dx (f[x]h[x]g[x]) = + ǫ ǫ R dxf[x]δ[x] dxf[x]δ[x] dxf[x]δ[x] R R − R R dxf[x]h[x]δ[x] Rd (f[x]g[x]δ[x]) dxfR[x]h[x]δ[x] dxδ[x]Dx (f[x]g[x]) ǫ 2 + ǫ 2 − R dxf[xR]δ[x] R dxfR[x]δ[x]

Dropping the boundaryR terms, we are left with: R

h[0] g[0]h′[0] (8.17) −

This looks like the Taylor series expansion around the unperturbed result.

Perturbative Expansion for Averages

Schematically, the restricted average can be written as: Chapter 8: Averaging Techniques and Consequences at Late Times 125

ddx det[g] det gµν∂ Q ∂ Q δ[Q ]S S = | | | µ j ν k| i i (8.18) h iQ1,...,Qn ddx det[g] det gµν∂ Q ∂ Q δ[Q ] R p | p| | µ j ν k|Q i i ddx(µ + µ + µ )(σ + σ + σ ) R p0 1 p2 0 1 2 Q (8.19) ddx(µ + µ + µ ) ≡R 0 1 2 R The above expression serves as a deﬁnition for the perturbative expansion of equation

(8.10). Nothing has been assumed about the coordinate system being used or the form of the averaging being described. While this notation is motivated by the one used in

[20], it is not the same. σ0, σ1 and σ2 is simply the series expansion for the observable which is being averaged. µ0, µ1 and µ2 can be found from the deﬁnition of the average.

Using the deﬁnition, we ﬁnd:

S =σ0 + σ1 + σ2

(0) µ0 =µ0:0 δ[Qi ] i Y (0) µ1 =µ1:0 δ[Qi ] i Y (1) ∂ (0) + µ0:0 Qj (0) δ[Qi ] j ∂Qj i X Y µ µ δµ 1:0 ≡ 0:0 µ =µ δµ δ[Q(0)] ⇒ 1 0:0 i i Y (1) ∂ (0) + µ0:0 Qj (0) δ[Qi ] j ∂Qj i X Y Chapter 8: Averaging Techniques and Consequences at Late Times 126

(0) µ2 =µ2:0 δ[Qi ] i Y (2) ∂ (0) + µ0:0 Qj (0) δ[Qi ] j ∂Qj i X Y (1) ∂ (0) + µ0:0δµ Qj (0) δ[Qi ] j ∂Qj i X Y 1 ∂2 + µ Q(1)Q(1) δ[Q(0)] 2 0:0 j k (0) (0) i ∂Qj ∂Q i Xj,k k Y µ µ δ(2)µ 2:0 ≡ 0:0 µ =µ δ(2)µ δ[Q(0)] ⇒ 2 0:0 i i Y (2) ∂ (0) + µ0:0 Qj (0) δ[Qi ] j ∂Qj i X Y (1) ∂ (0) + µ0:0δµ Qj (0) δ[Qi ] j ∂Qj i X Y 1 ∂2 + µ Q(1)Q(1) δ[Q(0)] (8.20) 2 0:0 j k (0) (0) i ∂Qj ∂Q i Xj,k k Y Performing the perturbative expansion, we ﬁnd that:

d d d d d d xµ σ d xµ σ d xµ σ d xµ1 d xµ0σ0 S = 0 0 + 0 1 + 1 0 h iQ1,...,Qn ddxµ ddxµ ddxµ − d 2 R 0 R 0 R 0 R d xµR 0 d d d d R d xµ0σ2 R d xµ1σ1 R d xµ1 d xµ0σ1 + + R ddxµ ddxµ − d 2 R 0 R 0 R d xµR 0 d 2 d d d R d xµ1 dR xµ0σ0 d xµR1 dxµ1σ0 + d 3 − d 2 R d xµR 0 R d xµR 0 d d d d xµ σ d xµ2 d xµ0σ0 + 2R 0 R (8.21) ddxµ − d 2 R 0 R d xµR 0 The above expressionR for a restricted geometricR average is completely general. In order to make progress, we need to be more speciﬁc. First, we deﬁne the average on the zeroth order manifold. We can write: Chapter 8: Averaging Techniques and Consequences at Late Times 127

ddxµ δ[Q(0)]X X = 0:0 i i (8.22) h i0 d (0) R d xµ0:0Q i δ[Qi ] R Q If an observable is dependent only on the constraints (S = S [Qi]), then that observable is a constant on the subspace and will commute with averaging. As an example, we consider the average matter density as an observable in a perfect FLRW universe, with a subspace deﬁned by a time dependent constraint. In this case, the subspace is a constant time hypersurface, and since the matter density is only time dependent, it will be the same everywhere on the hypersurface. As such, the observable commutes with averaging.

While the above statement is true, it is not useful beyond zeroth order aside from serving as a consistency check. In what follows we assume that the zeroth order observable is only dependent on the zeroth order constraints. Using the definition of the zeroth order average, the expression for the geometric average, and the perturbative definitions (equations (8.22,8.21,8.20)), we find that: Chapter 8: Averaging Techniques and Consequences at Late Times 128

(1) ∂ (2) ∂ S Q ,...,Q =σ0 + σ1 Qj σ0 + σ2 Qj σ0 h i 1 n − ∂Q(0) − ∂Q(0) * j +0 * j +0

(1) ∂ (1) ∂ (1) ∂ + Qk ln[µ0:0] Qk ln[µ0:0] σ1 Qj σ0 ∂Q(0) − ∂Q(0) − ∂Q(0) * * j +0 j ! j !+0

(1) ∂ + (δµ δµ 0) σ1 Qj σ0 −h i − ∂Q(0) * j !+0

∂ (1) ∂ (1) (1) ∂ + Qk Qk σ1 Qj σ0 ∂Q(0) − ∂Q(0) − ∂Q(0) * * k +0 k ! j !+0

(1) ∂ (1) ∂ Qk σ1 Qj σ0 − ∂Q(0) − ∂Q(0) * k j !+0 2 1 (1) (1) ∂ + Qj Qk σ0 (8.23) 2 ∂Q(0)∂Q(0) * j k +0 Similarly, the expression for the variance can be written as:

∆2S S2 S 2 ≡ Q1,...,Qn −h iQ1,...,Qn 2 (1) ∂ (1) ∂ = σ1 Qj σ0 σ1 Qj σ0 (8.24) − ∂Q(0) − − ∂Q(0) * j * j +0! +0

Spatial Averages of First Order Quantities on the Unperturbed Subspace

In general, we must consider the average of a ﬁrst order quantity on the unperturbed subspace. Using a longitudinal gauge as was described in section 6.1, we ﬁnd a correction of the form:

dηd3xf[η]δ[g[η]]ψ[η, ~x] Ψ = 0 dηd3xf[η]δ[g[η]] h i R R Chapter 8: Averaging Techniques and Consequences at Late Times 129

Here, f[η] is some zeroth order function, and therefore independent of ~x in an FLRW

universe. This expression can be written as:

d3xΨ[η, ~x] Ψ = h i0 d3x R g[η]=0 3 3 i~k ~x Rd k d xe · = ψ[η, k]E[~k] (2π)3/2 d3x Z R d3k δ[~k] = ψ[η, k]E[~k] (2π)3/2 δ[~0] Z ψ[η, 0]E[~0] = δ[~0](2π)3/2

We see that this expression is zero, provided that ψ[η, k]E[~k] is regular at ~k = ~0. Even

if this is not the case, the only time that terms of this type can have a contribution is

in the IR. If we consider the product of the spatial average of two ﬁrst order quantities,

once again we ﬁnd that there is zero contribution except for, perhaps, in the IR.

2 3 3 3 − d kd k′ 3 3 i~k ~x+ik~′ ~y Υ Ψ d x ψ[η, k]υ[η, k′]E[~k ]E[~k] d xd ye · · h i0 h i0 ≈ (2π)3 ′ Z Z Z 2 3 3 − d k 3 3 i~k (~x ~y) = d x ψ[η, k]υ[η, k] d xd ye · − (2π)3 | | Z 1 Z Z − d3k = d3x ψ[η, k]υ[η, k] d3xeikr cos[θ] (2π)3 | | Z Z 1 Z − k2dkr2dr = r2dr sin[kr] ψ[η, k]υ[η, k] 2π2kr | | Z Z n¯ 1 If we assume that ψ[η, k]υ[η, k] 2π2k3A¯ k − wheren ¯ = 1 corresponds to a scale | | ∼ k0 invariant power spectrum, we ﬁnd that: Chapter 8: Averaging Techniques and Consequences at Late Times 130

n¯ 1 R R 1 ∞ dk k − sin[kr] − Υ Ψ A¯ drr2 drr2 h i0 h i0 ≈ k k kr ZkIR 0 Z0 Z0

The spatial integral has been deﬁned on the domain r 0,R and the Fourier ∈ { } integral on the domain k k , . Taking the limits k 0 and R , this ∈ { IR ∞} IR → → ∞ expression is then dependent on kIRR, which is, in principle, unconstrained. If we place the spatial hypersurface into an inﬁnite spherical box with radius R we must have kR>π or kIRR = π. In this case we have:

0n> ¯ 1  Υ Ψ = A¯(1 π2cosintegral[π]) 0 0  − 2 n¯ =1 h i h i  π  n¯ 1 −  k − 3(¯n 3)tn 5 sin[πt] ¯ IR ∞ − A k 1 dt π3 n<¯ 1  0  n 5  ∞ 3(1R n¯)(¯n 3)t − sin[πt] ΥΨ dt − − ≤h i0 π3 Z1 Forn ¯ =0.963 we ﬁnd that:

ΥΨ Υ Ψ < h i0 h i0 h i0 900

All this to conclude that the product of two spatial averages can only be sensitive to

the long wavelength ﬂuctuations and, if it is, then the product of spatial averages is

much smaller than the spatial average of the product. In what follows we will choose

to ignore the spatial average of ﬁrst order quantities. Chapter 8: Averaging Techniques and Consequences at Late Times 131

One can argue that the appearance of terms sensitive to long wavelength ﬂuctuations

emerge as a result of an inconsistency in determining the background equations21.

The above expressions apply to any averaging that includes spatial averaging. As such, it applies to both spatial averaging and combined spatial and light cone averaging. It will not apply light cone averaging as discussed in [76].

In the case that spatial averaging is included in the formalism, we have:

(2) ∂ S Q ,...,Q σ0 + σ2 Qj σ0 h i 1 n ≈ − ∂Q(0) * j +0

(1) ∂ (1) ∂ + δµ Qk ln[µ0:0] σ1 Qj σ0 − ∂Q(0) − ∂Q(0) * k ! j !+0

∂ (1) (1) ∂ Qk σ1 Qj σ0 − ∂Q(0) − ∂Q(0) * k j !!+0 2 1 (1) (1) ∂ + Qj Qk σ0 (8.25) 2 ∂Q(0)∂Q(0) * j k +0 Similarly, the expression for the variance can be written as:

2 2 (1) ∂ ∆ S σ1 Qj σ0 (8.26) ≈ − ∂Q(0) * j ! +0

8.7.4 Adiabatic and Entropy Clocks

(0) (1) (2) (0) If we choose any two local observables ( = a + a + a and = + Oa O O O Ob Ob (1) + (2), we can define the relative entropy between them as22: Ob Ob 21See chapter 9. See also [72] for a discussion on absorbing IR divergences by redefining the background. 22This definition is closely tied to the usual definition of entropy perturbations [35], it has been extended to second order and will be applied not just to energy densities but to any arbitrary observable. Chapter 8: Averaging Techniques and Consequences at Late Times 132

(1) (1) (2) (2) (1) (1) ¨(0) ¨(0) a b a b a b a b Sab = O O + O O + O O O O ˙(0) − ˙(0) ˙(0) − ˙(0) ˙(0) ˙(0)  ˙(0) − ˙(0)  a a 2 a a O Ob O Ob O Ob O Ob   If we measure the average of as observed by all observers who see , and there Oa Ob is no relative entropy between them (Sab = 0), the average is unperturbed and backreaction plays no role:

(0) a a = a = a hO iO hO iOb O

In order for there to be zero entropy between two observables, we can write:

= [ ] Oa Oa Ob (0) + (1) + (2) = (0) + (1) + (2) Oa Oa Oa Oa Ob Ob Ob h d i d = [ (0)]+ (1) [ (0)]+ (2) [ (0)] Oa Ob Ob (0) Oa Ob Ob (0) Oa Ob d b d b 2 O O (1) 2 Ob d (0) + a[ ] (0)2 b 2 d O O Ob d (1) = (1) (0) ⇒Oa Ob (0) Oa d b O 2 (1) d b d2 (2) = (2) (0) + O (0) Oa Ob (0) Oa 2 (0)2 Oa d b d O Ob

The relative entropy between the two observables can be thought of as a measure of independence between the two observables. Chapter 8: Averaging Techniques and Consequences at Late Times 133

In this work, we consider three particular types of clocks. First, we consider the time as measured by geodesic observers (Buchert time). We will also consider observers who measure the same energy density of the matter ﬁeld. Finally, we consider observers who measure the same temperature of the CMB. This choice is motivated by the wealth of data from observing the anisotropies in the CMB. Chapter 9

Averaged Local Observables

In this chapter and the next, we will apply techniques developed in chapter 8 to both local and non-local expressions. This chapter will be devoted to the study of local observables, while chapter 10 will be devoted to studying the averaged expression for the luminosity distance as a function of the redshift.

The calculations are performed using two diﬀerent techniques, ensemble averaging, and averaging a hypersurface deﬁned by a particular clock. The clocks we consider here are the temperature of the CMB, Buchert time and the matter density.

The calculation is performed using a Mathematica script, which could be used to calculate the spatial average of any perturbative expression using any clock up to second order.

When considering spatial averaging using a clock, this script comes with a consistency check, namely performing the average over some observable using the same observable as a clock should lead to recovering the unperturbed FLRW result.

We begin by ﬁnding expressions for the averaged matter density, and then discuss

134 Chapter 9: Averaged Local Observables 135

the matter power spectrum and its sensitivity to long wavelength ﬂuctuations. The

average curvature of the submanifold on which local averaging is deﬁned can then be

interpreted as the spatial average of the universe.

When considering an ideal FLRW universe, one often considers the small redshift

expansion:

1 1 q0 2 dL = z + − z H0 2H0 Here the expansion rate and deceleration parameters are deﬁned by:

H U µ ≡∇µ U µ H q 1 ∇µ ≡− − H2

Using this formalism, we might test:

? 1 1 q d [χ ,z] = z + − z2 (9.1) h Liχ,LC 0 H 2H χ χ In the above expression, we perform the calculation up to quadratic order in the redshift (z), and χ refers to the clock used to deﬁne spatial averaging, and χ = χ0 picks

1 1 q − out the spatial hypersurface at current times. H χ and 2H χ are then calculated in terms of the average matter density, cosmological consta nt and averaged spatial

curvature.

As a toy model, the equality of equation (9.1) is assumed in order to ﬁnd corrections

to the spatial curvature and cosmological constant. Chapter 9: Averaged Local Observables 136

Finally we will summarize the results and observations from the chapter.

9.1 Energy Densities

The averaged energy densities can be written as1:

ρmat (0) 1+∆ensρ ρmat ≡

(2) =1 + δmat ρ h matiτ 1+∆ ρ (0) ≡ τ ρmat 3κ Λ =1 + + IP [2] (0) − 3Ω 10ρmat ρ h matirad 1+∆ ρ (0) ≡ rad ρmat 3δ(2) 3IP [0] =1 + δ(2) rad + 2IP [2] mat − 4 2 −

ρmat mat h (0)i 1+∆matρ ρmat ≡

(2) =1 + δmat

In each case we have introduced a shorthand for the correction. We notice that ∆χρ is a dimensionless correction to the averaged energy density where the average is performed using χ as a clock2, except for the case where spatial averaging has been

1When using the matter field as a clock, we use the expression given in equation (6.9), with (2) δmat = 0. This is done so that the background energy density of the matter field can be shifted in a way consistent with the shift which can be performed using the other clocks. If this were not the case, we would find that ∆matρ defined below vanishes. 2χ = τ corresponds to performing spatial averaging via Buchert averaging, χ = rad corresponds to using the temperature of the CMB as a clock, χ = mat corresponds to using the energy density of the matter field as a clock Chapter 9: Averaged Local Observables 137 performed via ensemble averaging. In the above expression, we have made use of the shorthand given in section 6.2:

dk k n IP [n]= P [k] k H Z 0 The cosmological constant commutes with averaging. In order to be consistent in our notation, we can deﬁne it in terms of the shorthand as:

Λ h iχ ∆ Λ (0) ≡ χ ρmat Λ = (0) ρmat

This is true for all clocks.

9.1.1 Integrated Power Spectra and IR Sensitivity

We can also deﬁne the integrated matter power spectrum in a nonperturbative way using:

2 ρmat ρmat χ −h i χ Pχρ = ρ 2 h matiχ dk = P ρ[k] k χ Z Using this deﬁnition, we can write: Chapter 9: Averaged Local Observables 138

8IP [2] 4IP [4] P ρ =4IP [0] + + end 3 9 4 P ρ = IP [4] τ 9 4IP [2] 4IP [4] P ρ =IP [0] + rad − 3 9

Pmatρ =0

Since there is no relative entropy between matter and itself, the fact that the integrated matter power spectrum is zero when using the matter density of the universe as a clock is not surprising. If, however, we use either of the other two clocks considered here, we do ﬁnd a nonzero contribution to the matter power spectrum. Since these two clocks have zero relative entropy to each other in the short wavelength limit, they will give a similar looking power spectrum in the UV. However, since they have nonzero relative entropy in the long wavelength limit, they will give rise to diﬀerent looking power spectra in the IR.

Locally, long wavelength fluctuations look like coordinate transformations on the background [147]. These fluctuations only appear globally when we consider a nonzero relative entropy between the clock defining observers and the observable in the long wavelength limit. If there is a relative entropy between the clock and the parameter, then the observable cannot be written as a function of the clock field alone ( = [χ]). O 6 O Since the longest wavelength fluctuations in an FLRW universe look homogeneous and isotropic, the appearance of long wavelength sensitivity implies that there is a need for the background solution to be adjusted, i.e. = [τ] or χ = χ[τ]. There must O 6 O 6 be some other parameter(s) in the background solution, which were not included. Chapter 9: Averaged Local Observables 139

It is worth noting that the metric ﬂuctuation is not a physical observable (it has gauge dependence). As such, it can be sensitive to long wavelength ﬂuctuations, while not giving rise to any physical observables that are sensitive to long wavelength

ﬂuctuations.

In our particular situation, we have added a radiation field under the assumption that it is too weak to affect the geometry, but strong enough to be detected and used as a clock. Alternatively stated, we assume we can make a radiation field that does not couple to gravity. This field gives rise to a global long wavelength sensitivity and, as such, is incorrect. If, however, one were to properly account for the radiation field, we should find that radiation and matter do not interact in the long wavelength limit3.

This would give rise to adiabatic long wavelength ﬂuctuations, which would in turn imply that there would be no global long wavelength sensitivity.

To summarize, locally IR divergences correspond to a gauge transformation. Glob- ally, IR divergences in physical observables imply a long wavelength relative entropy between the clock and the observable. This implies that the homogeneous, isotropic solution diﬀers from the background solution, or rather, that the background solution is incorrect. This is in agreement with [103, 79, 72]. See also [158, 148, 81] for discussions on using entropy perturbations to generate IR sensitive observables.

9.2 Ergodicity

In [75], the authors deﬁne an ergodic system as one which obeys:

3If they do interact in the long wavelength limit, then they cannot be treated as two perfect ﬂuids in the background FLRW universe. Chapter 9: Averaged Local Observables 140

F = F h i

Here we have deﬁne F as some arbitrary function of the matter density

F [ρmat[~x1],ρmat[~x2],... ].

In this analysis, we see that, in general, performing spatial averaging using the different clocks considered here leads to diﬀerent solutions, none of which are generally equal. This might lead one to ask the question, “What clock is needed in order to ensemble averaging to be equivalent to spatial averaging?”. We answer this question in two parts. First we perform a perturbative search for a clock for which ensemble averaging is equivalent to spatial averaging, and ﬁnd that no such clock exists. We then search for a clock for which our system is ergodic.

We begin with the assumption that we can deﬁne a clock ﬁeld which leads to spatial averages, which are equivalent to ensemble averages:

(0) (1) (1) χens = χens[η]+ χens[η, ~x]+ χens[η]

(0) (1) (2) Given any observable = a [η]+ a [η, ~x]+ a [η], we should ﬁnd that: Oa O O O

a = a hO iχens O 2 2 ( a) = ( a) O χens O

The analysis is easiest to begin with the deﬁnition of the variance. Up to second order, we must have that: Chapter 9: Averaged Local Observables 141

2 2 a a = a a O − hO iχens O − O χens Using equation (8.26), we ﬁnd that:

∂ χ(1) [η, ~x] (0) =0 ens ∂χ(0)[η]Oa

χ(1) [η, ~x] =0 ⇒ ens ∂χ(0)[η] or inf ∂η →

(1) The second solution is not useful and, as such, we have χens[η, ~x] = 0. At second order we must ﬁnd that:

= hOaiχens Oa (2) ∂ (0) (1) χens[η] (0) a =δµ a ∂χens[η]O O 1 (0) (0) − (2) (1) ∂χens ∂ a χens[η]=δµ a O ⇒ O ∂η ! ∂η !

This should be true for any arbitrary observables, which can only be true if all observables obey:

1 (0) − ∂ a (1) O a [η, ~x]= c ∂η ! O

This requirement is that all possible observables have zero relative entropy up to

first order. An example of two observables that do not have this property in the Chapter 9: Averaged Local Observables 142 spacetime defined in chapter 6 are ρ , as well as 2ρ . Since we can find at mat ∇ mat least two observables with nonzero relative entropy, we cannot find a submanifold of spacetime on which spatial averaging and ensemble averaging are the same operation.

In order to ensure ergodicity is achieved, one can deﬁne a clock:

1 (0) (0) − (2) (1) ∂χens ∂ a χens[η]=δµρmat O ∂η ! ∂η ! f [η] f [η]η3(∂ Φ [~x])2 χ [η]=f[η] ′ Φ [~x]2 ′ i 0 ⇒ ens − 3 0 − 36

Here f is an arbitrary function of η alone.

This function has been carefully constructed and, in general, does not correspond to any easily interpreted physical observable. In particular, if we were to solve the

Einstein equation in another gauge, we would find that the clock needed to ensure ergodicity would be different to the clock used here. In general, a clock which ensures ergodicity is not gauge invariant. Performing spatial averaging via ensemble averaging is therefore ill defined.

One might then question the use of ensemble averaging in general. We argue that any possible solution to the Einstein equation must obey4:

∂ 0= Φ[η, ~x] ∂η 3 d k i~k ~x Φ[η, ~x]= e · φ [k]E[~k] ⇒ (2π)3/2 0 Z

4We have ignored the decaying solution Chapter 9: Averaged Local Observables 143

The only assumption contained in this statement is that the Fourier transform of

the metric perturbation exists. One might then question the validity of the following

assumptions:

E[~k] =0

~ ~ 3 ~ ~ E[k]E[k′]=δ [k + k′]

Any fluctuations that obey these assumptions are Gaussian5. In particular, fluctuations with different wavenumbers are uncorrelated.

9.3 Scalar Spatial Curvature

Since we are considering a perturbed universe, the idea of spatial ﬂatness seems less important. If we choose a spatial hypersurface of the geometry speciﬁed by the clock

(χ), we might consider the curvature of the hypersurface as a deﬁnition for the spatial

curvature. Embeddings were discussed in section (8.2). In what follows, the covariant

derivative on the subspace is deﬁned by:

(h) t = ∂ t (h)Γc t ∇a b a b − ab c 1 where (h)Γc = hcd (∂ h + ∂ h ∂ h ) ab 2 a bd b ad − d ab Here, we have introduced the connection on the subspace, which can in turn be used

to calculate the curvature of the subspace. Since ensemble averaging does not have

5See [8] for current bounds on non Gaussianity in cosmology Chapter 9: Averaged Local Observables 144

a well-defined clock on which spatial averaging can be defined, we will simply define

the spatial average as the spatial average of the ensemble averaged metric. We ﬁnd

that the scalar curvature of the space can be written as6:

2 (3) 4 (0) Rens =144κη ρmat

2 2 2 (3) 4 (0) 960∂ Φ0[~x] 2720(∂iΦ0[~x]) Rτ =144κη ρmat 2 2 − 4 (0) − 4 (0) η ρmat η ρmat 2 2 2 (3) 4 (0) 1152∂ Φ0[~x] 3456(∂iΦ0[~x]) Rrad =144κη ρmat 2 2 − 4 (0) − 4 (0) η ρmat η ρmat 2 2 4 (3) 4 (0) 960∂ Φ0[~x] 32∂ Φ0[~x] Rmat =144κη ρmat 2 + 2 − 4 (0) 2 (0) η ρmat η ρmat 2 2 2 2 2 2720(∂iΦ0[~x]) 416(∂ Φ0[~x]) 8(∂ ∂iΦ0[~x]) 2 2 2 − 4 (0) − 2 (0) − (0) η ρmat 3η ρmat 3 ρmat Averaging these expressions, we ﬁnd that:

6These expressions are consistent with the Gauss-Codazzi equation (see lecture notes [31] as well as work on the ADM formalism [83]). Chapter 9: Averaged Local Observables 145

(3)R ens ∆ κ (0) ≡ ens ρmat κ = (0) ρmat (3)R h τ iτ ∆ κ (0) ≡ τ ρmat κ 10IP [2] = + (0) 3 ρmat (3) Rrad rad h (0) i ∆radκ ρmat ≡ κ 16IP [2] = (0) 3 ρmat − (3)R h matimat ∆ κ (0) ≡ mat ρmat κ 32IP [4] 8IP [6] = 10IP [2] (0) − − 9 − 27 ρmat We can clearly see that the submanifold deﬁned by observers with constant energy density is much more curved than the submanifold deﬁned by observers with constant radiation density, or constant Buchert time.

Since matter has a greater tendency to clump on smaller scales, it seems reasonable to ﬁnd that the curvature of the subspace of all observers at the same matter density is dominated by modes that are sensitive to short wavelength ﬂuctuations.

9.4 Expansion and Deceleration

The averaged expression for the expansion and deceleration parameters can be written in terms of the averaged energy density, cosmological constant and curvature. We

ﬁnd that: Chapter 9: Averaged Local Observables 146

1 3 ∆ κ ∆ ρ ∆ Λ 3IP [0] 121IP [2] 4IP [4] = 1+ ens ens ens + + H (0) 4 − 2 − 2 2 − 54 81 sρmat 1 3 ∆ κ ∆ ρ ∆ Λ 4IP [4] = 1+ τ τ τ + H (0) 4 − 2 − 2 81 τ sρmat 1 3 ∆ κ ∆ ρ ∆ Λ 3IP [0] 35 4IP [4] = 1+ rad rad rad + + IP [2] + H (0) 4 − 2 − 2 8 54 81 rad sρmat 1 3 ∆ κ ∆ ρ ∆ Λ 5IP [4] 2IP [6] = 1+ mat mat mat + + (9.2) H (0) 4 − 2 − 2 6 27 mat sρmat Similarly, for the deceleration parameter:

1 q 3 ∆ κ ∆ ρ 5∆ Λ 3IP [0] 229IP [2] 68IP [4] − = 1 ens ens + ens + + 2H (0) 1/2 − 4 − 2 2 2 54 − 81 4(3ρmat) 1 q 3 ∆ κ ∆ ρ 5∆ Λ 68IP [4] − = 1 τ τ + τ 2H (0) 1/2 − 4 − 2 2 − 81 τ 4(3ρmat) 1 q 3 ∆ κ ∆ ρ 5∆ Λ 3IP [0] 89 68IP [4] − = 1 rad mat + mat + IP [2] 2H (0) 1/2 − 4 − 2 2 8 − 54 − 81 rad 4(3ρmat) 1 q 3 ∆ κ ∆ ρ 5∆ Λ 317IP [4] 2IP [6] − = 1 mat mat + mat 2H (0) 1/2 − 4 − 2 2 − 162 − 27 mat 4(3ρmat) (9.3)

Aside from the discussion of sensitivity to long wavelength ﬂuctuations, equations (9.2,9.3)

are the main results of this section.

As a toy model, one can assume that the equality in equation (9.1) holds. If the

luminosity distance was ﬁtted accurately to the unperturbed model, we could, in

principle, solve the above expressions for the average energy density, curvature, and

dark energy density of the universe.

At this point the analysis is sensitive to the clock being used. If we consider the Chapter 9: Averaged Local Observables 147

energy density as being well deﬁned from some other measurement such as CMB

data, we ﬁnd that:

188IP [2] 16IP [4] ∆ κ = 9IP [0] + + ens − 27 27 3IP [0] 32 ∆ Λ= IP [2] + IP [4] ens − 2 − 81 16IP [4] ∆ κ = τ 27 32IP [4] ∆ Λ= τ 81 9IP [0] 43IP [2] 16IP [4] ∆ κ = + rad − 4 − 27 27 3IP [0] IP [2] 32IP [4] ∆ Λ= + rad − 8 − 2 81 179IP [4] 8IP [6] ∆ κ = mat − 81 − 27 91IP [4] ∆ Λ= mat 162

Based on this toy model, we might incorrectly conclude that short wavelength ﬂuc-

tuations cause the dark energy content of the universe to be overestimated and that

long wavelength ﬂuctuations cause the cosmological constant to be underestimated.

We might also correctly conclude that the spatial curvature is sensitive to the clock

which is used to deﬁne spatial hypersurfaces. In the case that the energy density of

the matter ﬁeld is used as a clock, we would be forced to conclude that the result is

dominated by the curvature of the averaging surface. Since IP [6] > 1, this result is beyond the validity of perturbation theory, and as such it does not apply. In chapter

11, we try to ﬁt the geometry described in chapter 6 to CMB data, as well as to supernova data. Chapter 9: Averaged Local Observables 148

9.5 Results

The main results of this chapter are as follows:

Global long wavelength sensitivity corresponds to an inconsistency in the back- • ground FLRW solution.

The use of the matter ﬁeld as a clock leads to a large nonperturbative correction, • which comes from the correction to the spatial curvature of the universe.

If equation (9.1) holds, then backreaction leads to an underestimation of the • spatial curvature and the cosmological constant. In chapter 10, we show that

this equation does not hold. Chapter 10

Averaging Nonlocal Observables

In this chapter we focus on ﬁnding the averaged expression for the luminosity distance of the universe by using the techniques described in chapter 8. Once this expression is found, we will write it in terms of the correction to the average spatial curvature, the averaged matter density, and a cosmological constant. This expression will then be used to test the small redshift expansion of the luminosity distance (see equation equation (9.1)).

Once again, the calculation is performed using a Mathematica script. In this case, the script calculates the combined spatial average for any observable using any clock to deﬁne spatial averaging. This calculation is only performed up to second order in perturbation theory.

Finally, we summarize the results of this chapter.

149 Chapter 10: Averaging Nonlocal Observables 150

10.1 Fractional Correction to the Area Distance

The luminosity distance is given by:

2 dL = (1+ z) dA

d =(1 + z)2 d ⇒h Li h Ai d d h Li =h Ai ⇒ (0) (0) dL dA

In the ﬁrst line, we have made use of Ethrington’s reciprocity theorem[71], and in the second line, we have noted that on constant redshift hypersurfaces, the factors of the redshift commute with averaging. In chapter 7, we found an expression for the area distance. The last line states that the fractional correction to the area distance is equal to the fractional correction to the luminosity distance. In this section, we quote the full expression for the correction to the luminosity distance. The expression is written in terms of the average matter density, averaged spatial curvature, and cosmological constant as found in chapter 9. We have also made use of the shorthand introduced

12 1 in section 6.2, where r = δη = (0) √1+z 1 : ρmat − Chapter 10: Averaging Nonlocal Observables 151

dk k 2 IP [n]= P [k] k φ H Z 0 dk k 2 IP [n,r]= P [k] sin[kr] sin k φ H Z 0 dk k 2 IP [n,r]= P [k] cos[kr] cos k φ H Z 0 dk k 2 IP [n,r]= P [k] sinintegral[kr] sinintegral k φ H Z 0 dk k 2 IP [n,r]= P [k] sin[kr]sinintegral[kr] sin,sinintegral k φ H Z 0 dk k 2 IP [n,r]= P [k] sin[kr]sinintegral[kr] cos,sinintegral k φ H Z 0

In the last line we have introduced the deﬁnitions IPsin,sinintegral[n,r] and

IPcos,sinintegral[n,r] are needed for the full expression. In the following pages we quote the full expression for the luminosity distance redshift relation. In the sections following this we will examine the terms that can be of importance. We begin with the expression for the luminosity distance after it has been averaged over the past light cone of a particular observer. Spatial averaging is performed via ensemble averaging, in the longitudinal gauge (see section 6.1): Chapter 10: Averaging Nonlocal Observables 152

d 1 h AiLC =1 ∆ ρ (0) − 2 ens dA r6 14r5 + 84r4 280r3 + 560r2 672r + 448 − − − ∆ Λ − 896 ens r2 +2r 4 − ∆ κ − 16 ens 4(r + 2)IP [ 2,r] sin,sinintegral − − r3 (3r 2)(r 2)3IP [1, 2r] + − − sin 36r3 2(r 2)2IP [0,r] + − sin,sinintegral 3r2 (9r2 7r + 2)(12r2 r + 2) IP [ 2, 2r] − − cos − − 9r6 (3r 2)(21r2 8r +4)(r 2)IP [ 1, 2r] − − − sin − − 18r5 (27r2 26r +8)(r 2)2IP [0, 2r] + − − cos 36r4 1 + r3IP [3,r] 10 sinintegral 2(11r + 4)IP [ 1,r] + sinintegral − r2 4(r2 +6r 5) IP [1,r] + − sinintegral 3r 416IP [ 2,r] + cos − 15r2 (93r4 270r3 + 320r2 360r + 400) IP [1,r] + − − sin 180r3 (999r4 270r3 1040r2 240r + 800) IP [0,r] + − − − cos 180r4 (3r4 + 15r3 70r2 + 100r 40) IP [2,r] + − − cos 30r2 (1017r4 30r3 380r2 60r + 200) IP [ 1,r] − − − sin − − 45r5 IP [0] (24r3 + 35r2 17r + 2) + − 9r3 IP [2] (59r4 282r3 + 386r2 336r + 64) + − − 108r2 IP [ 2] (708r4 + 465r3 245r2 + 80r 20) − − − (10.1) − 45r6 Chapter 10: Averaging Nonlocal Observables 153

If we perform the combined spatial and light cone averaging using Buchert time as a clock for deﬁning spatial averages, we ﬁnd the fractional correction to the area distance is given by:

dA 1 h iτ,LC =1 ∆ ρ (0) − 2 τ dA r6 14r5 + 84r4 280r3 + 560r2 672r + 448 − − − ∆ Λ − 896 τ r2 +2r 4 − ∆ κ − 16 τ 1 + r3IP [3,r] 10 sinintegral 2(r2 + 14r 6) IP [1,r] + − sinintegral 3r 416IP [ 2,r] + cos − 15r2 (3r4 + 370r3 1040r2 + 760r 80) IP [1,r] + − − sin 180r3 (9r4 + 15r3 70r2 + 100r 40) IP [2,r] + − − cos 90r2 (591r4 6130r3 + 4760r2 1520r + 160) IP [0,r] − − cos − 180r4 (1113r4 1465r3 + 1130r2 380r + 40) IP [ 1,r] + − − sin − 45r5 16IP [ 1,r] sinintegral − − 3r IP [0] (27r2 + 18r 16) − − 12r2 416IP [ 2] − − 15r2 IP [2] (73r4 94r3 + 440r2 544r + 128) + − − (10.2) 216r2 Chapter 10: Averaging Nonlocal Observables 154

If we perform the combined spatial and light cone averaging using the CMB temperature as a clock for deﬁning spatial averages, we ﬁnd the fractional correction to the area distance is given by:

dA 1 h irad,LC =1 ∆ ρ (0) − 2 rad dA r6 14r5 + 84r4 280r3 + 560r2 672r + 448 − − − ∆ Λ − 896 rad r2 +2r 4 − ∆ κ − 16 rad 1 + r3IP [3,r] 10 sinintegral 2(r2 + 14r 6) IP [1,r] + − sinintegral 3r 416IP [ 2,r] + cos − 15r2 (9r4 + 165r3 510r2 + 380r 40) IP [1,r] + − − sin 90r3 (9r4 + 15r3 70r2 + 100r 40) IP [2,r] + − − cos 90r2 (204r4 1725r3 + 1290r2 380r + 40) IP [0,r] − − cos − 45r4 (1548r4 1815r3 + 1230r2 380r + 40) IP [ 1,r] + − − sin − 45r5 12IP [ 1,r] sinintegral − − r 416IP [ 2] − − 15r2 IP [0] (39r2 48r 32) − − − 24r2 IP [2] (11r4 + 82r3 167r2 + 136r 32) − − (10.3) − 54r2 Chapter 10: Averaging Nonlocal Observables 155

If we perform the combined spatial and light cone averaging using the density of the matter field as a clock for defining spatial averages, we find the fractional correction to the area distance is given by: Chapter 10: Averaging Nonlocal Observables 156

dA 1 h imat,LC =1 ∆ ρ (0) − 2 mat dA r6 14r5 + 84r4 280r3 + 560r2 672r + 448 − − − ∆ Λ − 896 mat r2 +2r 4 − ∆ κ − 16 mat 1 + r3IP [3,r] 10 sinintegral 2(3r2 + 42r 38) IP [1,r] + − sinintegral 9r 416IP [ 2,r] + cos − 15r2 (9r4 + 1110r3 + 360r2 520r + 560) IP [1,r] + − sin 540r3 (27r4 + 45r3 435r2 + 1070r 520) IP [2,r] + − − cos 270r2 (591r4 6130r3 + 4760r2 1520r + 160) IP [0,r] − − cos − 180r4 (1113r4 1465r3 + 1130r2 380r + 40) IP [ 1,r] + − − sin − 45r5 16IP [ 1,r] sinintegral − − 3r (r 2)(3r 2)IP [3,r] + − − sin 54r 1 + IP [4] 12r2 24r + 73 54 − − 1 + IP [6] r2 2r +4 54 − − IP [0] (27r2 + 18r 16) − − 12r2 416IP [ 2] − − 15r2 IP [2] (107r4 + 454r3 860r2 + 40r 128) − − (10.4) − 216r2 Chapter 10: Averaging Nonlocal Observables 157

The variance of the area distance (divided by the area distance squared) where the average is taken over the past light cone and spatial averaging is done via ensemble averaging, is given by:

2 2 2 (dA dA ) (r 2) (3r 2) IP [0, 2r] −h iLC LC = − − cos (0) 2 − 72r4 (dA ) 2(r 2)(3r 2)IP [ 1,r] − − cos,sinintegral − − 3r3 (9r2 7r + 2)2 IP [ 2, 2r] + − cos − 18r6 (r 2)(9r2 7r + 2)(3r 2)IP [ 1, 2r] + − − − sin − 18r5 4(9r2 7r + 2) IP [ 2,r] + − sin,sinintegral − 3r4 1 + r3IP [3,r] 15 sinintegral (r2 + 14r 8) IP [1,r] + − sinintegral 3r 2(r 2)IP [ 1,r] − sinintegral − − r2 1 + r2IP [2,r] 15 cos 224IP [ 2,r] + cos − 15r2 (3r4 + 45r3 150r2 + 140r 40) IP [1,r] + − − sin 45r3 (81r4 765r3 + 780r2 340r + 80) IP [0,r] − − cos − 45r4 (627r4 945r3 + 780r2 340r + 80) IP [ 1,r] + − − sin − 45r5 IP [2](3r 2)2 (r2 4r + 8) + − − 108r2 IP [0] (183r4 + 24r3 + 16r2 64r + 16) − − 72r4 IP [ 2] (1749r4 630r3 + 425r2 140r + 20) − − − (10.5) − 90r6 Chapter 10: Averaging Nonlocal Observables 158

The variance of the area distance (divided by the area distance squared) where combined spatial light cone averaging is deﬁned using Buchert time as a clock is given by:

2 dA dA τ,LC −h i τ,LC 1 3 = r IPsinintegral[3,r] (0) 2 15 (dA ) (r2 + 14r 8) IP [1,r] + − sinintegral 3r 1 + r2IP [2,r] 15 cos 224IP [ 2,r] + cos − 15r2 (3r4 + 45r3 150r2 + 140r 40) IP [1,r] + − − sin 45r3 (117r4 1320r3 + 1260r2 560r + 160) IP [0,r] − − cos − 90r4 2(246r4 285r3 + 270r2 140r + 40) IP [ 1,r] + − − sin − 45r5 IP [2](3r 2)2 (r2 4r + 8) + − − 108r2 IP [0] (13r2 + 12r 12) − − 6r2 224IP [ 2] − (10.6) − 15r2 Chapter 10: Averaging Nonlocal Observables 159

The variance of the area distance (divided by the area distance squared) where combined spatial light cone averaging is deﬁned using the temperature of the CMB as a clock is given by:

2 dA dA rad,LC −h i rad,LC 1 3 = r IPsinintegral[3,r] (0) 2 15 (dA ) (r2 + 14r 8) IP [1,r] + − sinintegral 3r 1 + r2IP [2,r] 15 cos 224IP [ 2,r] + cos − 15r2 (3r4 + 45r3 150r2 + 140r 40) IP [1,r] + − − sin 45r3 (81r4 720r3 + 660r2 280r + 80) IP [0,r] − − cos − 45r4 (627r4 675r3 + 570r2 280r + 80) IP [ 1,r] + − − sin − 45r5 2IP [ 1,r] sinintegral − − r IP [2](3r 2)2 (r2 4r + 8) + − − 108r2 IP [0] (29r2 + 12r 24) − − 12r2 224IP [ 2] − (10.7) − 15r2 Chapter 10: Averaging Nonlocal Observables 160

The variance of the area distance (divided by the area distance squared) where combined spatial light cone averaging is deﬁned using the matter density as a clock is given by:

2 dA dA mat,LC −h i mat,LC 1 3 = r IPsinintegral[3,r] (0) 2 15 (dA ) (r2 + 14r 12) IP [1,r] + − sinintegral 3r 224IP [ 2,r] + cos − 15r2 (3r4 15r2 + 40r 20) IP [2,r] + − − cos 45r2 (3r4 + 45r3 60r2 + 70r 20) IP [1,r] + − − sin 45r3 (117r4 1320r3 + 1260r2 560r + 160) IP [0,r] − − cos − 90r4 2(246r4 285r3 + 270r2 140r + 40) IP [ 1,r] + − − sin − 45r5 IP [0] (13r2 + 12r 12) − − 6r2 224IP [ 2] − − 15r2 IP [2](9r4 48r3 + 88r2 40r + 32) + − − 108r2 IP [4] + (10.8) 9 Chapter 10: Averaging Nonlocal Observables 161

In order to test the results quoted on the previous pages, we make use of the consistency checks set out below.

No unphysical integration constants remain in the expression. In particular, the • constant which can be added to the expression for the radiation energy density

as well as the constant c1 which appear in equation (6.10) do not appear in any

ﬁnal results.

The homogeneous isotropic limit gives the correct expression. Explicitly, when • we take the limit that Φ[~x] 0, we recover the FLRW expression for the lumi- → nosity distance1. This is a nontrivial test since the expression for the luminosity

distance up to second order given in equation (7.4) does not contain any infor-

mation about the cosmological constant or the spatial curvature, and diﬀerent

clock ﬁelds have diﬀerent functional dependence on each of the cosmological

constant spatial curvature and correction to the matter density.

If we average any of the constraints, we ﬁnd that any perturbative corrections • vanish exactly. Since we are performing the average of the constraints over the

submanifold deﬁned by the constraint, this result is as expected.

The luminosity and the area distance are both zero at zero redshift. The area • distance approaches zero as the redshift approaches inﬁnity.

Although it is not a consistency check, we ﬁnd that there is no sensitivity to long • wavelength ﬂuctuations, except for the case that a nonadiabatic clock is used.

1The cosmological constant and the curvature are only included perturbatively at second order in our expression. If we compare the expression for the luminosity distance in a FLRW universe up to ﬁrst order in a cosmological constant and spatial curvature, we ﬁnd the results are in agreement. Chapter 10: Averaging Nonlocal Observables 162

This result is consistent with the ﬁndings of section 9.1.1. This is nontrivial

since the expression for area distance given in equation (7.4) is sensitive the

long wavelength ﬂuctuations.

10.2 Expansion and Deceleration

In chapter 9, we found the right hand side of the following relation:

? 1 1 q d [χ ,z ] = z + − z2 (10.9) h Liχ,LC 0 0 H 0 2H 0 χ χ Using the equation (10.2), and the corresponding expressions using the CMB temperature and matter density as clocks, we can calculate the left hand side:

3 ∆ κ ∆ ρ ∆ Λ 14IP [2] 4IP [4] d [τ,z ]= 1+ τ τ τ + + z h Liτ,LC 0 (0) 4 − 2 − 2 27 15 sρmat 3 ∆ κ ∆ ρ 5∆ Λ 38IP [2] 164IP [4] + 1 τ τ + τ z2 (0) 1/2 − 4 − 2 2 − 27 − 45 4(3ρmat) 3 ∆ κ ∆ ρ ∆ Λ 4IP [4] = 1+ τ τ τ + z 6 (0) 4 − 2 − 2 81 sρmat 3 ∆ κ ∆ ρ 5∆ Λ 68IP [4] + 1 τ τ + τ z2 (0) 1/2 − 4 − 2 2 − 81 4(3ρmat) 1 1 q = z + − z2 (10.10) H 0 2H 0 τ τ We see that in general:

1 1 q d [χ ,z ] = z + − z2 + ... (10.11) h Liχ,LC 0 0 6 H 0 2H 0 χ χ Chapter 10: Averaging Nonlocal Observables 163

For completeness, we can show that:

1 1 q d z − z2 h LiLC − H − 2H 38IP [2] 16IP [4] = 2IP [0] + + z − 27 81 IP [0] 77IP [2] 428IP [4] + + z2 2 54 − 405 1 1 q d z + − z2 h Liτ,LC − H 2H τ τ 14IP [2] 88IP [4] = + z 27 405 19IP [2] 284IP [4] + z2 − 54 − 405 1 1 q d z + − z2 h Lirad,LC − H 2H rad rad 16IP [2] 88IP [4] = 2IP [0] + + z 9 405 IP [0] 29IP [2] 284IP [4] + z2 2 − 18 − 405 1 1 q d z + − z2 h Limat,LC − H 2H mat mat 98IP [2] 548IP [4] = + z 27 405 67IP [2] 148IP [4] + z2 − 54 − 135

10.3 Large Redshift

In [138], Scrimgeour and collaborators argue homogeneity only applies on scales &

1 1 100h− Mpc. This corresponds to redshifts z & 30 . We could argue that for z . 0.03, Chapter 10: Averaging Nonlocal Observables 164

homogeneity does not apply, and any results could be diﬃcult to interpret in the

context of an FLRW universe.

H 2 For large redshifts (z > 10− ), we can make the approximations given in keq ∼ equations (6.17), (6.18), and (6.19).

We search for two particular types of terms. First we ﬁnd terms that are sensitive

to the IR cut oﬀ2. Aside from possible IR divergences, we argued in section6.2 that terms proportional to IP [n 2] and IP [n 2,x] π IP [n] give the largest ≥ sinintegral ≥ ≈ 2

H0 2 contributions for z > 10− . keq ∼ Using this approximation, we ﬁnd that the average fractional correction to the area

distance redshift relation can be approximated as:

2 The contribution from long wavelength ﬂuctuations are suppressed except for k H0, where k is in Fourier space. In searching for terms that are sensitive to the long wavelength ﬂuctuations,≪ we only search for terms that have IR divergences, and ignore terms that are IR safe. Chapter 10: Averaging Nonlocal Observables 165

dA 1 h iτ,LC 1 ∆ ρ (0) ≈ − 2 τ dA 3 z +2 + ∆τ κ 4√1+ z − 4(1 + z) z2 +3z +3 z3 +4z2 +6z +4 + ∆ Λ − 14(z + 1)5/2 14(z + 1)3 τ 36z 59 47z2 + 39z 126 + − − IP [2] 27√1+ z(√1+ z 1)2 − 54(1 + z)(√1+ z 1)2 − − π 2 3 + 2 IP [3] 20 − √1+ z 1 61 1 ∆ ρ IP [2] ≈ − 2 τ − 9 3 z +2 146 + ∆τ κ IP [2] 4√1+ z − 4(1 + z) − 27 z2 +3z +3 z3 +4z2 +6z +4 + ∆ Λ − 14(z + 1)5/2 14(z + 1)3 τ 2(107z + 137) 10(z + 3) + 2 2 IP [2] 27√z +1 √z +1 1 − 3 √z +1 1 ! − − π 2 3 + 2 IP [3] (10.12) 20 − √1+ z 2 d d A A τ,LC 2 −h i τ,LC 4(z + 2) 4z + 21z + 26 IP [2] (d(0))2 ≈ 9√1+ z(√1+ z 1)2 − 27(1 + z)(√1+ z 1)2 A − − π 2 3 + 2 IP [3] (10.13) 30 − √1+ z When studying the approximation, some terms have the same functional dependence as the average fractional correction to the matter density and spatial curvature. These terms cannot give rise to any eﬀects that look like anything other than spatial curvature or matter. We have therefore grouped all terms with the same functional form together. Terms which look like a shift in spatial curvature are grouped with spatial Chapter 10: Averaging Nonlocal Observables 166

curvature and terms which look like a shift in the matter density are grouped with

the matter density. Aside from the terms that can be accounted for as simple shifts

in the average energy density and average spatial curvature, there are terms that are

not as obvious when it comes to ﬁtting the data. The term proportional to IP [2]

is dominated by redshift space distortions, and the term proportional to IP [3] is a lensing correction to the area distance.

For redshifts 0.03 . z . 0.3, the term proportional to IP [2] dominates. In particular,

1 3 there is a z2 dependence for small z. We can identify this term as coming from :

(1) 2 ∂ log[µ0:0] ∂ log[σ0] 2(z + 2)(4z + 13) 8(z + 2) (z ) (0) (0) 2 2 IP [2] ∂z ∂z ≈ 27(z + 1) √z +1 1 − 9√z +1 √z +1 1 ! − − We note that there are other corrections proportional to IP [2] which emerge from a number of diﬀerent eﬀects, but none of them will be considered further here since they are dominated by either the redshift space distortions or the lensing term.

For redshifts z > 0.3, we ﬁnd a lensing correction that is proportional to IP [3], which ultimately emerges from terms containing two angular derivatives of the null parameter:

1 ηobs ∂2 + ∂2 dξΦ[ξ,r + η ξ,θ,ψ] θ sin2(θ) ψ source − Zηsource This is the usual lensing potential. Its eﬀect on variance has been studied previously

3 [140, 54, 55, 63, 33, 20]. Since IP [3] 10− , we would expect this lensing contribution ∼ to be important. This is discussed further in chapter 11.

3More speciﬁcally, we can identify this term as being proportional to the speed of the source relative to the observer squared. The low redshift contribution to the luminosity distance is clearly dominated by these redshift space distortions [97]. Chapter 10: Averaging Nonlocal Observables 167

In order to show the diﬀerences between the various expressions, it is instructive to

write:

d d h Airad,LC h Aiτ,LC (0) − (0) dA dA 1 19 20 ∆ ρ ∆ ρ IP [0] + IP [2] ≈− 2 rad − τ − 4 9 3 z +2 26 + ∆ensκ ∆τ κ + IP [2] 4√1+ z − 4(1 + z) − 3 2 2 dA dA rad,LC dA dA τ,LC −h i rad,LC −h i τ,LC (0) 2 − (0) 2 (dA ) (dA ) IP [0] ≈ 4

Based on this expression it seems that the effect of performing spatial averaging using the CMB as a clock, as opposed to using the Buchert averaging, leads to small changes in what an observer would consider the spatial curvature and the energy density of the universe. We cannot find any differences between these two clocks that look like anything other than a shift in the energy density or a shift in the spatial curvature. More interestingly, perhaps, is that long wavelength fluctuations cannot look like anything other than matter. This is in contrast to the findings of [113, 1], who find that long wavelength fluctuations look like a cosmological constant. This statement is always true provided that both the radiation field and the matter field are not interacting in the long wavelength limit. This can be seen as follows:

If the radiation field is not interacting with another field, the stress energy tensor of the radiation field must obey a continuity equation. In the long wavelength limit, one Chapter 10: Averaging Nonlocal Observables 168

has:

T µν =0 ∇µ rad ρ(0) ρ = rad (1 + δ [a[η]]) ⇒ rad a[η]4 rad

The function δrad[a[η]] must obey:

∂δ [a[η]] ∂φ[a[η]] rad = 4 ∂a[η] − ∂a[η]

Here the metric perturbation is introduced through the covariant derivative. We ﬁnd a similar expression for the matter density:

ρ(0) ρ = rad (1 + δ [a[η]]) mat a[η]3 mat ∂δ [a[η]] ∂φ[a[η]] mat = 3 ∂a[η] − ∂a[η]

These two equations have the general solution:

3δrad[a[η]]=4δmat[a[η]] + c

We therefore see that fluctuations in the radiation field look like an adiabatic fluctuation plus some constant. This constant could be interpreted as a correction to the background radiation field. Therefore, the effect of long wavelength fluctuations is to change the time at which a measurement was taken. Since our background only contains matter, the radiation field can only adjust the time of observation of the Chapter 10: Averaging Nonlocal Observables 169

background. It is therefore unsurprising that the IR corrections in our calculation

can only look like a change in the energy density of the universe

Next we consider the expressions found when using the matter ﬁeld as a clock.

d d h Aimat,LC h Aiτ,LC (0) − (0) dA dA 1 25 25 ∆ ρ ∆ ρ + IP [2] IP [4] ≈− 2 mat − τ 9 − 27 3 z +2 40 32 8 + ∆ensκ ∆τ κ + IP [2] + IP [4] + IP [6] 4√1+ z − 4(1 + z) − 3 9 27 7√z +1 + IP [2] 6(√z +1 1) − 2 2 dA dA mat,LC dA dA τ,LC −h i mat,LC −h i τ,LC (0) 2 − (0) 2 (dA ) (dA ) 1 IP [4] IP [2] + ≈3(√1+ z 1) 9 −

The ﬁrst thing to notice is the appearance of the term proportional to IP [6]. This particular term does not appear in the expression for the area distance until we write the area distance in terms of the averaged spatial curvature. It is not surprising that the curvature of the surface of constant energy density is so large, since matter has a

6 tendency to clump at late times. We recall that IP [6] evolves as H0− and, as such, the

eﬀect of this term is not as important at early times, especially before equal matter

and radiation.

We also notice the term proportional to IP [4], which in part arises from the deﬁnition of spatial curvature. These is however also a contribution which can be associated Chapter 10: Averaging Nonlocal Observables 170

with a shift in the matter density. Since we are considering all observers who see

a particular value for the energy density of the matter ﬁeld, there is no variance in

the energy density of the universe, and the variance which was previously associated

with the density of the matter ﬁeld emerges as a correction to the density of the

matter ﬁeld. This same variance can be seen in the varince of the area distance. The

corrections proportional to IP [2] are dominated by the corrections proportional to

IP [4] and IP [6] for all redshifts, and are therefore uninteresting.

For completeness, we can compare these results to those obtained when performing spatial averaging via ensemble averaging.

d dA h AiLC h iτ,LC (0) − (0) dA dA 1 128 ∆ ρ ∆ ρ + IP [0] IP [2] ≈− 2 ens − τ − 27 3 z +2 10 + ∆ensκ ∆τ κ IP [2] 4√1+ z − 4(1 + z) − − 3 4 37z + 70√z +1 72 + − − IP [2] 27 z √z +1+1 − ! 2 2 dA dA τ,LC (dA dA ) −h i τ,LC −h iLC LC (0) 2 − (0) 2 (dA ) (dA ) 0 ≈

We see that this expression contains an IR sensitive term that, as before, can be

absorbed into a shift in the energy density of the matter ﬁeld. We do, however, ﬁnd a

small diﬀerence between the two expressions which cannot be absorbed into the deﬁ- Chapter 10: Averaging Nonlocal Observables 171 nition of the scalar curvature or the energy density. This term is most important for small redshift, and is related to the backreaction between the clock and the redshift, in particular redshift space distortions.

10.4 Discussion

Having found expressions for the average fractional correction to the area distance

(equation (10.12)), as well as the variance associated with it (equation (10.13)), we are now in a position to discuss the cosmological implications. In what follows, we will only consider the expression found using Buchert time as a clock.

In [92], the authors correctly identified the dominant contributions to the variance as coming from redshift space distortions for low redshifts, as well as coming from weak lensing for high redshifts, a result which is consistent with our findings in spite, of the details of our calculation not matching the details of their calculation. In their work, they did not discuss the effect of backreaction corrections on the luminosity distance redshift relation, but only on its variance.

We will brieﬂy discuss the implications of the above ﬁndings. In general, the luminosity distance is related to the apparent magnitude (m) and the zero point magnitude

(M) of a standard candle by:

m = 5log10[dL]+ M

When ﬁtting supernova data to a model, we must account for this zero point magnitude. We usually make use of supernovae at low redshifts to calibrate the zero point Chapter 10: Averaging Nonlocal Observables 172 magnitude4. Once the zero point magnitude has been found, we can use low redshift supernovae to ﬁnd the slope of the luminosity distance redshift relation, which in turn allows us to calculate the energy content of the universe. Using higher redshift supernovae allows us to calculate the shape of the luminosity distance redshift relation, which in turn allows us to calculate the equation of state of the universe.

When studying low redshift supernovae, we must account for the following sources of error:

As with all measurements, we have a number of systematic errors, as well as an • intrinsic scatter in the supernova data. Since there are such a large number of

supernovae found at small redshifts (z < 0.3), the latter are less important.

Redshift space distortions will give rise to a scatter in the supernova data. If the • position of the supernova is recorded, the wealth of data from galaxy surveys

can help to understand this scatter.

At second order, there will be a nonzero average correction to the luminosity • distance redshift relation. Since these corrections are not usually accounted

for when ﬁtting supernova data to a model, these limitations could lead us to

misinterpret some of the features of the model. In particular, the slope of the

luminosity distance redshift relation is slightly increased. We might incorrectly

interpret such an observation as there being less matter in the universe. While

the correction for z 0.03 can be important, it dies off quickly, and we find ∼ that by z 0.1 there is almost no average effect on the luminosity distance due ∼

4This is closely related to the Cepheid calibrations discussed in [135]. Chapter 10: Averaging Nonlocal Observables 173

to redshift space distortions.

When studying high redshift supernovae, we must account for the following sources of error:

As was the case for low redshift supernovae, we have a number of systematic • errors, as well as an intrinsic scatter in the supernova data. There have been

less supernovae found at higher redshifts and, as such, the intrinsic scatter is

signiﬁcant.

Errors associated with the calibration of the zero point magnitude will also • propagate to the interpretation of large redshift supernovae.

Weak gravitational lensing will give rise to a scatter in the supernova data. •

At second order, there will be a nonzero average correction to luminosity dis- • tance redshift relation. Once again, these corrections are not usually accounted

for when ﬁtting supernova data to a model. The limitations of the model will

contribute to the misidentiﬁcation of certain features. We ﬁnd that, for large

redshifts, the luminosity distance is increased by gravitational lensing. The ef-

fect of gravitational lensing is therefore to lower the apparent equation of state

of the universe. Since this correction is proportional to IP [3], this eﬀect could

be signiﬁcant. As such, ﬁtting this term to various data will be the subject of

chapter 11. Chapter 11

Fitting the Data

In chapter 6 we constructed a matter dominated universe, with inhomogeneous corrections to the matter density, as well as homogeneous isotropic corrections to the spatial curvature, matter content as well as the cosmological constant. In chapter 9 we used the expressions from chapter 6 to ﬁnd expressions for the fractional correction to the average energy density, the average spatial curvature, and the cosmological constant in terms of the homogeneous and isotropic corrections to them, the backreaction of the ﬁrst order matter perturbations (which contained terms proportional to IP [0],

(0) IP [2], IP [4] and IP [6]), and the background parameter ρmat.

(0) Λ=ρmat∆χΛ

(3) (0) Rχ χ =ρmat∆χκ

(0) ρ =ρ (1+∆ ρ) h matiχ mat χ

In chapter 10 we found an expression for the luminosity distance in terms of the

174 Chapter 11: Fitting the Data 175

averaged matter density, spatial curvature and dark energy content of the universe.

(0) Since ∆χΛ,∆χκ,∆χρ, ρmat, IP [2], and IP [3] are not directly measurable, it is the focus of this chapter to understand the implications of ﬁtting cosmological data to this model including backreaction on the cosmological parameters.

The remainder of this chapter will consider the problem of ﬁtting a perturbative model to data. We then discuss the approximate size of backreaction, and then discuss the implications of both supernova data as well as CMB data.

11.1 Least Squares Fitting and Perturbation The-

ory

In general, we consider observables ( ) which depend on some independent variables O (x), to which our model or our free parameters (β) must be ﬁtted. To take one example, we could consider the luminosity distance as our observable, and the redshift as its independent variable. To this relation, we must ﬁt the free parameters of our model, namely: average energy density, average spatial curvature, cosmological constant, and backreaction. In what follows, we will write the dependence of the model on the free parameters as well as the independent variables as:

[β][x] O

Since we will in general have data relating to x, one can use a least squares proce- O dure to ﬁt the data. Chapter 11: Fitting the Data 176

N ( [β][x ] )2 χ2 = O i −Oi σ2 n=1 i X In the above expression, we have N data points, where takes on a value √σ O Oi ± i

when x takes a value xi. We assume no uncertainty in the independent variables (x),

and the uncertainty the observable is given by σi. The best ﬁt is found by searching

for the values of the free parameters which minimize χ2.

Now if we choose to build our model using perturbation theory in some of the free

parameters, care must be taken. To begin we will separate the background and the

perturbative parameters β = β0, β¯ , where β0 are the background parameters, and

β¯ are the perturbative parameters. One then has:

∂ [β , β¯] ∂2 [β , β¯] [β , β¯][x] [β , 0] + β¯ O 0 + β¯ β¯ O 0 O 0 ≈O 0 i ∂β¯ i j ∂β¯ ∂β¯ i β¯=0 i j β¯=0

In this case, we have:

N ¯ 2 1 2 ¯ ∂ [β0, β][xi] χ 2 ( [β0, 0][xi] i) 2 iβa O ≈ σ O −O − O ∂β¯ ¯ n=1 i a β=0 X ∂ [β , β¯][x ] ∂ [β , β¯][x ] ∂2 [β , β¯][x ] +β¯ β¯ O 0 i O 0 i 2 β¯ β¯ O 0 i a b ∂β¯ ∂β¯ − Oi a b ∂β¯ ∂β¯ a b β¯=0 a b β¯=0!

In this case, we have a quadratic function in β¯i which is minimised by:

N ¯ ¯ ¯ 2 ¯ 1 ∂ [β0, β][xi] ¯ ∂ [β0, β][xi] ∂ [β0, β][xi] ¯ ∂ [β0, β][xi] 0 2 i O + βb O O 2 iβb O ≈ σ O ∂β¯ ¯ ∂β¯ ∂β¯ ¯ − O ∂β¯ ∂β¯ ¯ n=1 i a β=0 a b β=0 a b β=0! X

Chapter 11: Fitting the Data 177

Our calculation is now dependent on the order to which perturbation theory is con-

sidered. If we ﬁnd a perturbative expression for up to linear order in β¯, we either O ignore the last term in the above expression, leading to inaccurate results, or we only perform the calculation of χ2 up to leading order in β¯, in which case the system is unbounded from below, and can’t be minimised.

Since the calculation of part II of this thesis are accurate only to the leading nonvanishing correction, performing any data fitting is poorly defined. As such, we can only estimate the size of effects, and speculate as to the implications of these effects.

11.2 Size of Backreaction

Before ﬁtting our expression to the data, it is worth ﬁrst considering the approximate size of the backreaction terms:

dk k n IP [n]= P [k] k H Z 0 keq k n A ≈ 0 H0 Z Z n A keq ≈ n H 0 9 2 n 10− 10 (11.1) ∼ In order to estimate the size of keq , we assume that k is approximately given by H0 eq the comoving Hubble rate at equal matter and radiation. We therefore have that

keq 102 a crude approximation, we have: H0 ∼ 3 From this, we can see that IP [3] 10− , IP [4] 0.1 and IP [6] > 1, but are ∼ ∼ Chapter 11: Fitting the Data 178 sensitive to the details of the power spectrum in the UV, and are therefore sensitive to nonlinear corrections, which are not as well understood. We will not assume any priors as to the size of these terms. IP [2], however, is not as sensitive to short

5 wavelength ﬂuctuations, and as such we ﬁnd IP [2] 10− . While there are large ≈ error bars associated with this term, it cannot be large enough to have a sizable impact on the expression for the area distance.

In what follows, the quoted CMB results are taken from [150], in which the authors consider CMB data from WMAP5 [105, 65, 88] and ACBAR [133] for l 40. In ≥ considering only the large multipole moments, they ﬁnd a number of model independent constraints on various cosmological parameters. In particular, they quote expressions for the area distance to the surface of last scattering, as well as the total baryonic and dark matter content of the universe. These parameters are quoted without any prior assumptions on the curvature or dark energy content of the universe.

They do, however, assume that the physics up to the surface of last scattering is well understood.

In order to ﬁnd the correction to the luminosity distance redshift relation, the two most important data points are the area distance to the surface of last scattering, and the total matter density of the universe.

Since we are considering matter and dark matter as a single ﬂuid in this analysis, we only need: Chapter 11: Fitting the Data 179

ρ =3H2Ω h mati mat 2 100 2 2 =3 h Ω Mpc− 3 105 mat × 8 2 =4.83(1 0.05) 10− Mpc− ± ×

The area distance to the surface of last scattering is given by:

D [z = z ]=12.7(1 0.02)Mpc h Ai ∗ ±

In order to change between luminosity distance and area distance, we need an expression for redshift of last scattering:

z∗ =1094(1 0.001) ±

Based on these parameters from the CMB and the variance in each, we might expect that corrections to the cosmological parameters will be quoted with variance 5%. ∼ Backreaction might lead to corrections of the form:

ρ ρ(0) (1 + α IP [n]) h matiτ ∼ mat n n=2 X In this case, we ﬁnd that any correction with α . 0.05 is consistent with the data. | n| IP [n] In particular, we ﬁnd that α . 103 is consistent with the data. The corrections α , | 2| n>2 however, are not as small and may have implications on cosmological parameters. Chapter 11: Fitting the Data 180

11.3 Supernova Data

Here we consider supernova from [143]. We fit the data in two different ways. First we consider that equations (10.12,10.13) are a good fit to the data. Next we consider a nonperturbative fit assuming an FLRW universe, and find a polynomial fit to the variance. Based on the assumption that the variance and the correction to the central value are approximately the same size, we reduce all the data points by some scaled factor of the variance, then perform a fit to this adjusted data, and find the differences in the parameters obtained. Each of these situations will be addressed in more detail in the following subsections.

11.3.1 Fitting Perturbative Expressions

In order to go about ﬁtting equations (10.12,10.13) to the supernova data we proceed as follows.

Fit ρ(0) to the data, so that the background is as close to FLRW as possible. • mat

Using ρ(0) , ﬁt ∆ρ, ∆κ, ∆Λ to the data in terms of IP [2] and IP [3]. • mat

Using ρ(0) , ∆ρ, ∆κ and ∆Λ, ﬁt IP [2] and IP [3] to the expression for the vari- • mat ance.

Figure 11.1 shows the results of this ﬁtting procedure. We see that allowing the spatial curvature to be nonzero allows for a better ﬁt, but a relatively large spatial curvature. We also see that making the prior assumption that spatial curvature is zero gives Λ > 1 which indicates a breakdown in perturbation theory. In each ρmat+Λ Chapter 11: Fitting the Data 181

L L -6 -6 : ® 0.304258, ® 2.12254, IPH2L ® 1.94205´10 , IPH3L ® 1.> : ® 2.7812, IPH2L ® -2.78474´10 , IPH3L ® 37.6353> < Ρm > +L < Ρm > +L < Ρm > +L d Mpc dLHMpcL LH L

12 000 12 000

10 000 10 000

8000 8000

6000 6000

4000 4000

2000 2000

z z 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 11.1: Figures showing the luminosity distance redshift relation, using the data

from [143] in blue. A best ﬁt line of the background to the data is shown in green.

Perturbative corrections are shown in red, and 1σ uncertainty associated with cosmic variance is shown in orange. The figure on the left allows for nonzero spatial curvature and the figure on the right makes the assumption the spatial curvature is zero. case we find that the data favors a value of IP [3] & 1. In each case this indicates that the perturbative approach is not well justified.

Next one might consider the expression for the cosmic variance. Figure 11.2 shows

2 (dL dL ) −h i 2 for the model allowing a nonzero spatial curvature as well as the model h dL i h i with zero spatial curvature.

11.3.2 Fitting Nonperturbative Expressions to estimate the

size of the Corrections

In this section we do not try to ﬁt equations (10.12,eq:fractionalvarareadistlargez) to

the supernova data. Instead, we ﬁt supernova data to an exact ﬂat FLRW universe

with a cosmological constant, and matter. We then estimate the size of the pertur- Chapter 11: Fitting the Data 182

2 2 < HdL - < dL >L > < HdL - < dL >L >

2 2 < dL > < dL >

2 2

1 1

z z 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-1 -1

Figure 11.2: Figures showing the cosmic variance in the luminosity distance as a function of redshift, using the data from [143] ﬁtted to equations

(10.12,eq:fractionalvarareadistlargez), both with (left) and without (right) spatial curvature. bative correction and how much impact that might have on cosmological parameters.

The procedure is as follows.

Fit h, Ω to the data. • m

Perform a polynomial ﬁt to the cosmic variance. Identify the variance associated • with cosmic variance and lensing.

Reduce the data points by some factor ( (1)) multiplied by the cosmic vari- • O ance1.

Fit h, Ω to the data after being scaled by cosmic variance. • m

Compare h and Ω before and after the data is reduced. • m

1In a matter dominated universe, equations (10.12,eq:fractionalvarareadistlargez) predict a factor 3 of 2 for redshift space distortions and a factor of 2 for lensing corrections. In what follows we generously perform the calculation with a factor of 10 and 100 respectively Chapter 11: Fitting the Data 183

WL : ® 0.642507, h ® 0.690376> WL + Wm dLHMpcL 12 000

10 000

8000

6000

4000

2000

z 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 11.3: Figure showing the luminosity distance redshift relation, using the data from [143] in blue. A best ﬁt line to the data is shown in red, and 1σ uncertainty associated with cosmic variance is shown in orange.

The best fit of a flat FLRW model to the data from [143] is shown in figure 11.3.

We can see that the cosmological parameters are given by the usual values. Figure

11.4 shows the approximate 68% confidence ellipse associated with the cosmological parameters when fitted to supernova data. Figure 11.5 shows a polynomial fit to the variance from the central value assuming a flat FLRW model. We can clearly see the redshift space distortions are important at low redshifts (z 0.05), and lensing ∼ correction at z & 0.6. We see that the data between z 0.1 and z 0.5 have minimal ∼ ∼ 6 cosmic variance. We also see that the this data is consistent with IP [2] 10− and ∼ 2 IP [3] 10− , which is consistent with the estimates of equation (11.1). ∼ In order to adjust the dataset, we perform the following change: Chapter 11: Fitting the Data 184

Wm 0.6

0.5

0.4

0.3

0.2

0.1 h 0.60 0.65 0.70 0.75 0.80

Figure 11.4: Figure showing the 68% conﬁdence ellipse and best ﬁt for the Hubble parameter and matter content of the universe, using supernova data alone.

2 < HdL - < dL >L >

2 < dL >

0.0014

0.0012

0.0010

0.0008

0.0006

0.0004

0.0002

z 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 11.5: Figure showing the cosmic variance in the luminosity distance as a function of redshift, using the data from [143] ﬁtted to a polynomial of order 5. Chapter 11: Fitting the Data 185

WL WL : ® 0.642339, h ® 0.690755> : ® 0.64094, h ® 0.694209> WL + Wm WL + Wm dLHMpcL dLHMpcL 12 000

10 000 10 000

8000 8000

6000 6000

4000 4000

2000 2000

z z 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 11.6: Figures showing the luminosity distance redshift relation, using the data

from [143] reduced by a factor of q (q = 10 left and q = 100 right) multiplied by the

cosmic variance. A best ﬁt line shown in red, and 1σ uncertainty associated with

cosmic variance is shown in orange.

2 (dL dL ) di di 1 q −h i L 7→ L − 2 dL i ! h i z=z

Here q is simply a scaling factor which relates the cosmic variance to the correction to the Luminosity distance. We generously take q = 10 and q = 100 in what follows.

The results are plotted in ﬁgure 11.6.

Based on cosmic variance estimates, one might expect a slight increase in the Hubble

constant ∆H0 . 0.5% and a slight decrease in the dark energy fraction of the universe H0 ∆ΩΛ . 0.25%. We see that supernova constraints on cosmological parameters might ΩΛ not be sensitive to the eﬀects of inhomogeneities and anisotropies. Chapter 11: Fitting the Data 186

11.4 Conclusions

The main results of part II of this thesis are set out below:

1. The leading correction to the area distance is similar in size to the zeroth or-

der expression, when using perturbation theory. This indicates a breakdown

in perturbation theory. When estimating the size of the perturbative correc-

tions to an FLRW solution, it seems that the perturbative corrections will be

much smaller. It is unclear as to whether or not perturbative corrections are

important, particularly at large redshifts when the lensing correction becomes

more important. An alternative approximation technique or a full numerical

calculation are needed before more concrete conclusions can be drawn.

2. Diﬀerent observers will see diﬀerent expressions for the luminosity distance red-

shift relation. For clocks that are sensitive to short wavelength ﬂuctuations in

the universe, such as the matter density, we ﬁnd that clock sensitive terms dom-

inate the luminosity distance redshift relation. The variance in the matter ﬁeld

is translated into large corrections to other observables. In this case, we also ﬁnd

that the curvature of the subspace of all similar observers is large. If, however,

we consider clocks that are not sensitive to the short wavelength ﬂuctuations,

such as the energy density of the radiation ﬁeld or the time as measured by

a geodesic observer, we ﬁnd that the clock sensitive terms are subdominant.

In particular, all observers who see the same temperature of the CMB will see

similar expressions for the luminosity distance redshift relation. If an entropy

ﬁeld is used as a clock, the IR sensitive correction looks like a change in the Chapter 11: Fitting the Data 187

matter content of the universe. As such, we cannot make any claims that long

wavelength ﬂuctuations mimic or mask dark energy.

3. Redshift space distortions dominate the perturbative correction to the luminos-

ity distance redshift relation at small redshift (z < 0.2). These redshift space

distortions will, on average, make it appear that there is extra energy density

in the universe. This eﬀect is small for redshifts above the homogeneity scale

(z & 0.03)[138]. While it will be measurable, it is likely to manifest as an un-

certainty in the measurement of the Hubble constant rather than having major

cosmological implications.

4. A lensing correction dominates at high redshifts (z > 0.5), but has little eﬀect

at low redshifts. It can do little to change the slope of the luminosity distance

redshift relation, but it might aﬀect the shape in a similar way to dark energy.

5. In general, the slope of the luminosity distance redshift relation is not inversely

proportional to the expansion rate of the universe. Neither of these expressions

is related to the slope of the luminosity distance redshift relation above the scale

of spatial homogeneity (z > 0.03)[138]. In essence, local considerations of the

expansion rate of the universe cannot be trusted.

6. The best ﬁt to the data including backreaction might require a slightly larger

Hubble constant and a slightly smaller dark energy fraction than assuming

backreaction does not contribute. It is possible that the size of this correction to

the various cosmological parameters is signiﬁcantly smaller than the uncertainty

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